■初等関数研究村■

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分からない問題はここに書いてね456
リーマン予想証明終了

1 ：2019/06/15 ～最終レス：2020/04/15: 初等関数（しょとうかんすう、英: Elementary function）とは、
実数または複素数の1変数関数で、代数関数、指数関数、対数関数、
三角関数、逆三角関数および、それらの合成関数を作ることを
有限回繰り返して得られる関数のことである

ガンマ関数、楕円関数、ベッセル関数、誤差関数などは初等関数でない
初等関数のうちで代数関数でないものを初等超越関数という
双曲線関数やその逆関数も初等関数である

初等関数の導関数はつねに初等関数になる
2 ：: 縦3マス、横4マスの12マスのうちランダムに選ばれた
2マスにそれぞれ宝が眠っている
AEIBFJ…の順で縦に宝を探していく方法をとるP君と、
ABCDEFGH…の順で横に宝を探していく方法をとるQ君が、
同時に地点Aから探索を開始した
どっちの方が有利？

ABCD
EFGH
I JK L

P1st Q1st even
[1,] 0 0 1
[2,] 4 5 6
[3,] 26 27 13
[4,] 84 83 23
[5,] 203 197 35
[6,] 413 398 50
[7,] 751 722 67
[8,] 1259 1210 87
[9,] 1986 1910 109
[10,] 2986 2875 134

完全追尾型多項式が完成しました

宝の個数は2

P1st={12n^4+28n^3-42n^2-52n-3(-1)^n+51}/48

Q1st={12n^4+20n^3-18n^2-20n-3(-1)^n+3}/48

even={10n^2+8n+(-1)^n-9}/8

■Wolframに入力すると既約分数表示になるので御注意

P1st／Q1st

=8(n-1){(n-2)n-6}/{2n(n+2)(6n^2-2n-5)-3(-1)^n+3}+1
3 ：: P1stとQ1stは、『宝一つの時の自陣当たり数』の二乗と
それぞれの差分を表す関数の和で求められる

■P1stを求める

宝一つの時の自陣当たり数

n(n+1)/2-1　……①

P1stは①^2と差分の和

差分は0 0 1 3 7 13 22 34 50 70 95 125 161 203
252 308 372 444 525 615……

それを表す関数

(4n^3-6n^2-4n-3(-1)^n+3)/48　……②

計算知能で①^2+②を入力すると

∴P1st={12n^4+28n^3-42n^2-52n-3(-1)^n+51}/48

■Q1stを求める

宝一つの時の自陣当たり数

n(n+1)/2-1　……①

Q1stは①^2と差分の和

差分は0 1 2 2 1 -2 -7 -15 -26 -41 -60 -84 -113
-148 -189……

それを表す関数は　

(-4n^3+18n^2+28n-3(-1)^n-45)/48　……③

計算知能で①^2+③を入力すると

∴Q1st={12n^4+20n^3-18n^2-20n-3(-1)^n+3}/48

■evenを求める

evenは、n(n+1)-1と同着数の和

同着数は1 2 4 6 9 12 16 20 25……

これを表す関数は　{2n^2-1+(-1)^(n)}/8　……④

n(n+1)-1　……⑤

計算知能で④+⑤を入力すると

∴even={10n^2+8n+(-1)^n-9}/8
4 ：: P1st Q1st even
[1,] 0 0 1
[2,] 4 5 6
[3,] 26 27 13
[4,] 84 83 23
[5,] 203 197 35
[6,] 413 398 50
[7,] 751 722 67
[8,] 1259 1210 87
[9,] 1986 1910 109
[10,] 2986 2875 134
[11,] 4320 4165 161
[12,] 6054 5845 191
[13,] 8261 7987 223
[14,] 11019 10668 258
[15,] 14413 13972 295
[16,] 18533 17988 335
[17,] 23476 22812 377
[18,] 29344 28545 422
[19,] 36246 35295 469
[20,] 44296 43175 519

Table[(12n^4+28n^3-42n^2-52n-3(-1)^n+51)/48,{n,1,20}]

Table[(12n^4+20n^3-18n^2-20n-3(-1)^n+3)/48,{n,1,20}]

Table[(10n^2+8n+(-1)^n-9)/8,{n,1,20}]
5 ：: 2×3の場合
宝：1個　同等
宝：2～3個　長軸有利
宝：4～6個　同等

□■■
□□■

短軸有利☆

Table[C(3,k-1)+C(1,k-1),{k,1,6}]
{2, 4, 3, 1, 0, 0}

長軸有利☆

Table[C(3,k-1)+C(2,k-1),{k,1,6}]
{2, 5, 4, 1, 0, 0}

同等☆

Table[C(5,k-1)+C(3,k-2)+C(1,k),{k,1,6}]
{2, 6, 13, 13, 6, 1}

2 * 3 [2] : 4 , 5 , 6
2 * 3 [3] : 3 , 4 , 13
6 ：: > sapply(1:12,function(k) treasure0(3,4,k))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12]
短軸有利 5 26 73 133 167 148 91 37 9 1 0 0
長軸有利 5 27 76 140 176 153 92 37 9 1 0 0
同等 2 13 71 222 449 623 609 421 202 64 12 1

□■■■
□□■■
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短軸有利☆

Table[sum[C(2n-1+C(0,n-2),k-1),{n,1,5}],{k,1,12}]

長軸有利☆

Table[sum[C(2n-1+C(0,3mod n),k-1),{n,1,5}],{k,1,12}]

同等☆

Table[C(11,k-1)+C(9,k-2)+C(7,k-2)+C(1,k),{k,1,12}]
7 ：: > sapply(1:20,function(k) treasure0(4,5,k))
[,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11]
短軸有利 9 84 463 1776 5076 11249 19797 28057 32243 30095 22749
長軸有利 9 83 453 1753 5075 11353 20057 28400 32528 30250 22803
同等 2 23 224 1316 5353 16158 37666 69513 103189 124411 122408
[,12] [,13] [,14] [,15] [,16] [,17] [,18] [,19] [,20]
短軸有利 13820 6656 2486 695 137 17 1 0 0
長軸有利 13831 6657 2486 695 137 17 1 0 0
同等 98319 64207 33788 14114 4571 1106 188 20 1

4×5の場合
宝：1個　同等
宝：2～5個　短軸有利
宝：6～13個　長軸有利
宝：14～20個　同等

□■■■■
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短軸有利☆

Table[sum[C(2n-1+C(0,n-2)+C(1,n-4),k-1),{n,1,9}],{k,1,20}]

長軸有利☆

Table[sum[C(2n-1+C(0,3 mod n)-C(0,n-5)+C(0,n-6),k-1),{n,1,9}],{k,1,20}]

同等☆

Table[C(19,k-1)+C(17,k-2)+C(15,k-2)+C(13,k-2)+C(8,k-2)+C(1,k),{k,1,20}]
8 ：: 5×6の場合
宝：1個　同等
宝：2～8個　短軸有利
宝：9～21個　長軸有利
宝：22～30個　同等

□■■■■■
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短軸有利☆

Table[sum[C(2n-1+C(0,n-2 mod7)+3C(0,n-4)+C(1,n-7),k-1),{n,1,14}],{k,1,30}]

長軸有利☆

Table[sum[C(2n-1+C(0,30mod n)-C(0,n-2)-2C(0,n-5)-C(1,n-8),k-1),{n,1,14}],{k,1,30}]

同等☆

Table[sum[C(2n-1-3C(1,n-9),k-2),{n,9,14}],{k,1,30}]+Table[C(29,k-1)+C(1,k),{k,1,30}]

5 * 6 [2] : 203 , 197 , 35
5 * 6 [3] : 1801 , 1727 , 532
5 * 6 [4] : 11418 , 11008 , 4979
5 * 6 [5] : 55469 , 54036 , 33001
5 * 6 [6] : 215265 , 211894 , 166616
5 * 6 [7] : 685784 , 680768 , 669248
5 * 6 [8] : 1827737 , 1825076 , 2200112
5 * 6 [9] : 4130886 , 4139080 , 6037184
5 * 6 [10] : 7995426 , 8023257 , 14026332
5 * 6 [11] : 13346984 , 13395944 , 27884372
5 * 6 [12] : 19312228 , 19372871 , 47808126
5 * 6 [13] : 24301031 , 24358063 , 71100756
5 * 6 [14] : 26642430 , 26684251 , 92095994
5 * 6 [15] : 25463979 , 25488051 , 104165490
9 ：: 6×7の場合
宝：1個　同等
宝：2～12個　短軸有利
宝：13～31個　長軸有利
宝：32～42個　同等

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短軸有利☆

Table[sum[C(2n-1+C(0,n-2)+3C(0,n-4)+5C(0,n-7)+C(1,n-11)+C(1,n-13),k-1),{n,1,20}],{k,1,42}]

長軸有利☆

Table[sum[C(2n-1+C(0,30mod n)-C(0,n-2 mod12)-2C(0,n-5)-3C(0,n-9)-C(1,n-12),k-1),{n,1,20}],{k,1,42}]

同等☆

Table[sum[C(2n-1-3C(1,n-14)-3C(0,n-13)-8C(0,n-12),k-2),{n,12,20}],{k,1,42}]+Table[C(41,k-1)+C(1,k),{k,1,42}]

6 * 7 [2] : 413 , 398 , 50
6 * 7 [3] : 5328 , 5070 , 1082
6 * 7 [4] : 49802 , 47536 , 14592
6 * 7 [5] : 361511 , 347863 , 141294
6 * 7 [6] : 2125414 , 2063677 , 1056695
6 * 7 [7] : 10409448 , 10191338 , 6377542
6 * 7 [8] : 43330401 , 42718984 , 31980800
6 * 7 [9] : 155608539 , 154251591 , 136031680
6 * 7 [10] : 487675145 , 485359843 , 498407985
6 * 7 [11] : 1345799489 , 1343074613 , 1591687274
6 * 7 [12] : 3293603485 , 3292560662 , 4471952741
6 * 7 [13] : 7189071864 , 7193592264 , 11136067152
6 * 7 [14] : 14059388483 , 14074085203 , 24726755394
6 * 7 [15] : 24725171790 , 24753058778 , 49194197048
6 * 7 [16] : 39214892052 , 39255073592 , 88039755958
6 * 7 [17] : 56218716543 , 56265877603 , 142177333010
6 * 7 [18] : 72972907098 , 73019303768 , 207704910184
6 * 7 [19] : 85862179541 , 85900953866 , 275012177393
6 * 7 [20] : 91643393740 , 91671084359 , 330477129321
6 * 7 [21] : 88747779232 , 88764701159 , 360745394049

10 ：: 7×8の場合
宝：1個　同等
宝：2～16個　短軸有利
宝：17～43個　長軸有利
宝：44～56個　同等

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短軸有利☆

Table[sum[C(2n-1+C(0,n-2 mod18)+3C(0,n-4)+3C(1,n-7)+7C(0,n-11)+C(1,n-16)+C(1,n-18),k-1),{n,1,27}],{k,1,56}]

長軸有利☆

Table[sum[C(2n-1+C(0,n-1 mod14)+C(0,n-3 mod18)+3C(1,n-5)+3C(1,n-9)-19C(0,n-14)-C(1,n-17)-C(1,n-19),k-1),{n,1,27}],{k,1,56}]

同等☆

Table[sum[C(2n-1-3C(1,n-20)-3C(1,n-18)-8C(1,n-16),k-2),{n,16,27}],{k,1,56}]+Table[C(55,k-1)+C(1,k),{k,1,56}]
11 ：: 7 * 8 [2] : 751 , 722 , 67
7 * 8 [3] : 13213 , 12546 , 1961
7 * 8 [4] : 169815 , 161494 , 35981
7 * 8 [5] : 1708176 , 1634573 , 477067
7 * 8 [6] : 14026034 , 13521709 , 4920693
7 * 8 [7] : 96716833 , 93921622 , 41278945
7 * 8 [8] : 571625198 , 558773693 , 290095184
7 * 8 [9] : 2940723248 , 2890925540 , 1744319612
7 * 8 [10] : 13327198939 , 13162957237 , 9116895304
7 * 8 [11] : 53717709609 , 53254225291 , 41930280380
7 * 8 [12] : 194070976396 , 192951568390 , 171360762514
7 * 8 [13] : 632475500322 , 630177011156 , 627260220922
7 * 8 [14] : 1869295969469 , 1865362789969 , 2070073204362
7 * 8 [15] : 5032748390589 , 5027434867987 , 6193066240064
7 * 8 [16] : 12389874719763 , 12385213035831 , 16873864084671
7 * 8 [17] : 27980641402960 , 27981556314178 , 42035336024662
7 * 8 [18] : 58125229289763 , 58139877526913 , 96062882957224
7 * 8 [19] : 111326498505381 , 111364943071921 , 201964537970498
7 * 8 [20] : 196977669970830 , 197048666795639 , 391587225396961
7 * 8 [21] : 322510102010304 , 322617018858127 , 701638985697449
7 * 8 [22] : 489306306855569 , 489444206271532 , 1163831929136799
7 * 8 [23] : 688690248074025 , 688846020744196 , 1789759515397979
7 * 8 [24] : 900050700996225 , 900206640621300 , 2554774361679750
7 * 8 [25] : 1092975958236546 , 1093115221856691 , 3388349400127275
7 * 8 [26] : 1233862233565383 , 1233973593552186 , 4178612556991503
7 * 8 [27] : 1295273249461927 , 1295353120172050 , 4794316279376103
7 * 8 [28] : 1264553645519991 , 1264605044607097 , 5119531910633352
12 ：: 宝一つの時の自陣当たり数

n(n+1)/2-1　

https://i.stack.imgur.com/3aEGX.png

大きな数字のところでは誤差があります

http://codepad.org/VN03aiqT
13 ：: 同等8 * 9 [18] : 14798849190259080
短軸8 * 9 [18] : 13325129660655316
長軸8 * 9 [18] : 13308110914669040

から誤差がある
14 ：: ■8x9マスで宝マックス72個テーブルも一瞬で表示

短軸有利☆

Table[sum[C(2n-1+C(0,n-2)+3C(0,n-4)+3C(1,n-7)+7C(0,n-11)+C(1,n-12)+9C(0,n-16)+C(1,n-22)+C(1,n-24)+C(1,n-26),k-1),{n,1,35}],{k,1,72}]

{35, 1259, 28901, 487245, 6460920, 70274262, 645084445, 5101533131, 35303844988,
216412209627, 1186682990705, 5867639936202, 26336848147168, 107913286582509,
405577089880106, 1403922286907797, 4491874681282838, 13325129660655319,
36749474808714593, 94449719219262517, 226689450187793573,
509035059085166018, 1071176160573816479, 2115432026610089700,
3925691963352022341, 6853294513073859630, 11266129211141121742,
17454698843693046407, 25505307844551837326, 35172169563389617239,
45797547548960471211, 56330082290098069195, 65468524173196415705,
71914624215592018826, 74671243825552686388, 73292765675007905651,
68001993326895424179, 59631707476231518911, 49411792162802982783,
38676208214646507895, 28584945063602478482, 19938274802884300793,
13116714709717265237, 8132639200776732766, 4748278261200713338,
2608024858933092322, 1346074794408997564, 652006213752455743,
295956138898867441, 125683998661458955, 49842381651879601,
18418955705334457, 6327555809439679, 2015233315978833,
593168628408153, 160782910480936, 39968340729272, 9068194179784,
1867271369048, 346638007264, 57550022756, 8461928362, 1088598639,
120646033, 11286483, 866713, 52461, 2347, 69, 1, 0, 0}
15 ：: ■8x9マスで宝マックス72個テーブルも一瞬で表示

同等☆

Table[sum[C(2n-1-3C(0,n-28)-3C(1,n-26)-3C(1,n-24)-8C(0,n-23)-8C(1,n-21)-15C(0,n-20),k-2),{n,20,35}],{k,1,72}]+Table[C(71,k-1)+C(1,k),{k,1,72}]

{2, 87, 3295, 78607, 1362299, 18460078, 204473689, 1907116083, 15299719813, 107274376311,
665613316422, 3691399441605, 18447776156424, 83642334863742, 346035607900560,
1312638938412806, 4584809892945575, 14798849190259082, 44283503920739404,
123188383908980963, 319353810087020272, 773186685811315639, 1751591017389233568,
3719181606403019809, 7412653767304185445, 13886128424486382893,
24477720915701752696, 40642683785697114854, 63620630278918684964,
93961096384315847204, 131013012205871839238, 172557237876989179559,
214781731322670114329, 252731141418076935138, 281209274772956576193,
295926350847761236653, 294548347126207473781, 277298087576831730532,
246896780442822393205, 207866926373152892934, 165440348653912344087,
124431016360680033348, 88399759656981333882, 59288415686663225877,
37514631338865127956, 22377473721141027910, 12572352774184755184,
6646249228402815124, 3302093433054131533, 1539874630017375451,
673008134822102446, 275211143609823985, 105099248767176058,
37401623133599593, 12373255757373154, 3794739201203181,
1075517359850959, 280687932668752, 67172923268624, 14670008286928,
2907185390840, 519288075532, 82935807842, 11727724279, 1450536738,
154505482, 13886622, 1024096, 59502, 2554, 72, 1}
16 ：: しかも誤差を修正済み

いやぁ、この出力は圧巻ですね
Haskell先生もびっくり
しかし誤差あり
17 ：: 宝箱問題、
もとの 4x3 型の12部屋で宝箱の数を変えてみると
1と8以上で有利不利無し、それ以外は長軸優先有利となるな
初見での印象よりも随分奥深いなこれ

計算式お願いする

プログラムで計算したので式はなんとも

4x5だと宝箱を増やすと途中で短軸有利から長軸有利に
変わっちゃうので自分でもびっくりした
18 ：: n=8くらいまでならマスのサイズを固定した場合、
宝を1からマックスまで変化させるロジックは完全に解明された
19 ：: □■■■■■■■■
□□★■■■■■■
□□□★■■■■■
□☆□□★■■■■
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□□☆□□□■■■
□□□□□□□■■
□□□☆□□□□■

{69, 67, 65, 63, 61, 59, 57, 56, 52, 50, 48, 46, 44, 43, 42, 37, 35, 33, 32, 31, 30, 24, 23, 22, 21, 20, 15, 14, 13, 12, 8, 7, 6, 3, 2}

35項目、合計1210

8x9マス長軸は三角数の位置2 6 12 20 30 42 56で1上がっている
つまり、最大マスから一回りづつ小さいマスの総数は全て数える

8x9マスでは8(8+1)/2-1=35　35項

>>4[8,] 1259 1210 87　から合計1210
20 ：: 8 * 9 [2] : 1259 , 1210 , 87
8 * 9 [3] : 28901 , 27444 , 3295
8 * 9 [4] : 487245 , 462938 , 78607
8 * 9 [5] : 6460920 , 6168325 , 1362299
8 * 9 [6] : 70274262 , 67504568 , 18460078
8 * 9 [7] : 645084445 , 623551570 , 204473689
8 * 9 [8] : 5101533131 , 4960367131 , 1907116083
8 * 9 [9] : 35303844988 , 34509440319 , 15299719813
8 * 9 [10] : 216412209627 , 212525346318 , 107274376311
8 * 9 [11] : 1186682990705 , 1169989129225 , 665613316422
8 * 9 [12] : 5867639936202 , 5804244923649 , 3691399441605
8 * 9 [13] : 26336848147168 , 26122841703128 , 18447776156424
8 * 9 [14] : 107913286582509 , 107268699582069 , 83642334863742
8 * 9 [15] : 405577089880106 , 403841343528838 , 346035607900560
8 * 9 [16] : 1403922286907797 , 1399743796844505 , 1312638938412806
8 * 9 [17] : 4491874681282838 , 4482908439962531 , 4584809892945575
8 * 9 [18] : 13325129660655316 , 13308110914669040 , 14798849190259080
8 * 9 [19] : 36749474808714576 , 36721381656941040 , 44283503920739408
8 * 9 [20] : 94449719219262544 , 94410951895703376 , 123188383908980944
8 * 9 [21] : 226689450187793600 , 226649637879721216 , 319353810087020288
8 * 9 [22] : 509035059085166144 , 509020882643576960 , 773186685811315328
8 * 9 [23] : 1071176160573816448 , 1071238534080555904 , 1751591017389233920
8 * 9 [24] : 2115432026610089728 , 2115648029075918592 , 3719181606403020288
8 * 9 [25] : 3925691963352023040 , 3926156660554725888 , 7412653767304184832
8 * 9 [26] : 6853294513073858560 , 6854100615782599680 , 13886128424486381568
8 * 9 [27] : 11266129211141124096 , 11267338149222707200 , 24477720915701743616
8 * 9 [28] : 17454698843693041664 , 17456312814286665728 , 40642683785697116160
8 * 9 [29] : 25505307844551831552 , 25507254963487424512 , 63620630278918684672
8 * 9 [30] : 35172169563389628416 , 35174310810267590656 , 93961096384315801600
21 ：: ■8x9マス長軸テーブル外せば出力可能

sum[C(2n-1+C(0,3mod n)+C(0,n-6 mod15)+C(0,n-10 mod18)+C(0,n-15)-C(0,n-5 mod22)-3C(0,n-9)-3C(1,n-13)-7C(0,n-20)-C(1,n-23)-C(1,n-25),k-1),{n,1,35}],k=16

1399743796844505

>>20
8 * 9 [16] : 1399743796844505
22 ：: k=26,　　　　6854100615782599621

8 * 9 [26] : 6854100615782599680
23 ：: Table[sum[C(2n-1+α,k-1),{n,1,a}],{k,1,b}]

a=n(n+1)/2-1　
b=n(n+1)

を満たす差分追尾数列αを見つけてくれ～(・ω・)ノ
24 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
43bDtYxhwTZApiRugaYbzsFfDKZmuR212sTlb3HrYDe9jytaLBgeojHWZxPkzLDqO3djDHR4YEE7wySKMFA1WfilujRqD7izkWmcUWhFiRZrxFJAByshrPMNynEJEpGtdOg7Qx1jjMcB2nGphazIOhgkUuKFgGIiMh65hqcNYbtLdSVeTIncn2bR8pUncW95wGWymgC0
J8hGG9KXTgycc8wu65xqHO4p5z5oqxLBQRZuY5NdFK6pM1UaaUzIAlBvvWS49LKCsiIbUDX0KKFIWjAdkRpo4aZkTjXtlBABqjBeDgeH64Kv8QRCkv4NklJWJU6uagXQG8uqws2ZLhzyEs04D6ycyNc9s3LAeDaywG9mQ3jlBFhHE7ba2qlDJlN9ixypMXRleRQZWryk
5f1wRmcZzXXh9xqvyHtHqtXG06b6iQ5OhfrAUZwU8Scwkh52X7iNRot4vwSfMrmjYoGlVIvhK9djdlkiGy03ly9O6SmKKfkBYZJCK8zLNCJux0nBGVJWVe90kIjRFBTCjOfe11bfeVXfLUM9mLp0zyFrfY4a1dC7nS9pShB2iDxRGp6Vn2SlReeXnc6mqJ6KfhY9L8gR
G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
25 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
43bDtYxhwTZApiRugaYbzsFfDKZmuR212sTlb3HrYDe9jytaLBgeojHWZxPkzLDqO3djDHR4YEE7wySKMFA1WfilujRqD7izkWmcUWhFiRZrxFJAByshrPMNynEJEpGtdOg7Qx1jjMcB2nGphazIOhgkUuKFgGIiMh65hqcNYbtLdSVeTIncn2bR8pUncW95wGWymgC0
J8hGG9KXTgycc8wu65xqHO4p5z5oqxLBQRZuY5NdFK6pM1UaaUzIAlBvvWS49LKCsiIbUDX0KKFIWjAdkRpo4aZkTjXtlBABqjBeDgeH64Kv8QRCkv4NklJWJU6uagXQG8uqws2ZLhzyEs04D6ycyNc9s3LAeDaywG9mQ3jlBFhHE7ba2qlDJlN9ixypMXRleRQZWryk
5f1wRmcZzXXh9xqvyHtHqtXG06b6iQ5OhfrAUZwU8Scwkh52X7iNRot4vwSfMrmjYoGlVIvhK9djdlkiGy03ly9O6SmKKfkBYZJCK8zLNCJux0nBGVJWVe90kIjRFBTCjOfe11bfeVXfLUM9mLp0zyFrfY4a1dC7nS9pShB2iDxRGp6Vn2SlReeXnc6mqJ6KfhY9L8gR
G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
26 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
43bDtYxhwTZApiRugaYbzsFfDKZmuR212sTlb3HrYDe9jytaLBgeojHWZxPkzLDqO3djDHR4YEE7wySKMFA1WfilujRqD7izkWmcUWhFiRZrxFJAByshrPMNynEJEpGtdOg7Qx1jjMcB2nGphazIOhgkUuKFgGIiMh65hqcNYbtLdSVeTIncn2bR8pUncW95wGWymgC0
J8hGG9KXTgycc8wu65xqHO4p5z5oqxLBQRZuY5NdFK6pM1UaaUzIAlBvvWS49LKCsiIbUDX0KKFIWjAdkRpo4aZkTjXtlBABqjBeDgeH64Kv8QRCkv4NklJWJU6uagXQG8uqws2ZLhzyEs04D6ycyNc9s3LAeDaywG9mQ3jlBFhHE7ba2qlDJlN9ixypMXRleRQZWryk
5f1wRmcZzXXh9xqvyHtHqtXG06b6iQ5OhfrAUZwU8Scwkh52X7iNRot4vwSfMrmjYoGlVIvhK9djdlkiGy03ly9O6SmKKfkBYZJCK8zLNCJux0nBGVJWVe90kIjRFBTCjOfe11bfeVXfLUM9mLp0zyFrfY4a1dC7nS9pShB2iDxRGp6Vn2SlReeXnc6mqJ6KfhY9L8gR
G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
27 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
43bDtYxhwTZApiRugaYbzsFfDKZmuR212sTlb3HrYDe9jytaLBgeojHWZxPkzLDqO3djDHR4YEE7wySKMFA1WfilujRqD7izkWmcUWhFiRZrxFJAByshrPMNynEJEpGtdOg7Qx1jjMcB2nGphazIOhgkUuKFgGIiMh65hqcNYbtLdSVeTIncn2bR8pUncW95wGWymgC0
J8hGG9KXTgycc8wu65xqHO4p5z5oqxLBQRZuY5NdFK6pM1UaaUzIAlBvvWS49LKCsiIbUDX0KKFIWjAdkRpo4aZkTjXtlBABqjBeDgeH64Kv8QRCkv4NklJWJU6uagXQG8uqws2ZLhzyEs04D6ycyNc9s3LAeDaywG9mQ3jlBFhHE7ba2qlDJlN9ixypMXRleRQZWryk
5f1wRmcZzXXh9xqvyHtHqtXG06b6iQ5OhfrAUZwU8Scwkh52X7iNRot4vwSfMrmjYoGlVIvhK9djdlkiGy03ly9O6SmKKfkBYZJCK8zLNCJux0nBGVJWVe90kIjRFBTCjOfe11bfeVXfLUM9mLp0zyFrfY4a1dC7nS9pShB2iDxRGp6Vn2SlReeXnc6mqJ6KfhY9L8gR
G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
28 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
43bDtYxhwTZApiRugaYbzsFfDKZmuR212sTlb3HrYDe9jytaLBgeojHWZxPkzLDqO3djDHR4YEE7wySKMFA1WfilujRqD7izkWmcUWhFiRZrxFJAByshrPMNynEJEpGtdOg7Qx1jjMcB2nGphazIOhgkUuKFgGIiMh65hqcNYbtLdSVeTIncn2bR8pUncW95wGWymgC0
J8hGG9KXTgycc8wu65xqHO4p5z5oqxLBQRZuY5NdFK6pM1UaaUzIAlBvvWS49LKCsiIbUDX0KKFIWjAdkRpo4aZkTjXtlBABqjBeDgeH64Kv8QRCkv4NklJWJU6uagXQG8uqws2ZLhzyEs04D6ycyNc9s3LAeDaywG9mQ3jlBFhHE7ba2qlDJlN9ixypMXRleRQZWryk
5f1wRmcZzXXh9xqvyHtHqtXG06b6iQ5OhfrAUZwU8Scwkh52X7iNRot4vwSfMrmjYoGlVIvhK9djdlkiGy03ly9O6SmKKfkBYZJCK8zLNCJux0nBGVJWVe90kIjRFBTCjOfe11bfeVXfLUM9mLp0zyFrfY4a1dC7nS9pShB2iDxRGp6Vn2SlReeXnc6mqJ6KfhY9L8gR
G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
29 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;l

30 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;l
31 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
32 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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G1SMJQYyXNSnrSkV2JyMHPsH4umT9610YnFgDQnLn67betRIHPmdDewhiu5kGTYKxytAxhC1qJPuOVRw5q7ZpELCBwYEixZs6kmHkla4fdAZ5HQw7xnySrJ28cGOloZuerh0QEG6xb3JxuzHxFGdBWtgEy5decyx9iZpMlo5QVT14hFNLnp0zlcAGUfEdxAMjUP3lHkv
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kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;l
33 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;l
34 ：: ※前スレ

■初等関数研究所■
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/
35 ：: 2 3 6 7 9
2 3 6 7 8 12 13 15 17
2 3 6 7 8 12 13 14 16 20 21 23 25 27
2 3 6 7 8 12 13 14 15 20 21 22 24 26 30 31 33 35 37 39
2 3 6 7 8 12 13 14 15 20 21 22 23 25 30 31 32 34 36 38 42 43 45 47 49 51 53

長軸choose数え上げ
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37 ：: 2n x 2n の正方形を
1 x 2 のドミノで埋める場合の数を考えます

たとえば、2x2の正方形を1x2のドミノで埋める場合の数は、2通りです

4x4の正方形を1x2のドミノで埋める場合の数は、36通りです

一般に、n=0,1,2,3,,,,のとき、
1, 2, 36, 6728, 12988816, 258584046368,,,
となり、一般項は、

Π[j=1 to n]Π[k=1 to n]{4cos^2 πj/(2n+1)+4cos^2 πk/(2n+1)}

となるようなのですが、
どのようにその公式が導かれるのでしょうか?
38 ：: wikipedia
https://en.wikipedia.org/wiki/Domino_tiling
によると
Temperley & Fisher (1961) and Kasteleyn (1961)
によって独立に発見されたとある
多分元論文は
Temperley, H. N. V.; Fisher, Michael E. (1961),
"Dimer problem in statistical mechanics-an exact result",
Philosophical Magazine, 6 (68): 1061-1063, doi:10.1080/14786436108243366

Kasteleyn, P. W. (1961), "The statistics of dimers on a lattice. I.
The number of dimer arrangements on a quadratic lattice", Physica,
27 (12): 1209-1225, Bibcode:1961Phy....27.1209K, doi:10.1016/0031-8914(61)90063-5.

原論文読むのが早い
39 ：: これに証明載ってるかも
https://inis.iaea.org/collection/NCLCollectionStore/_Public/38/098/38098203.pdf?r=1&r=1

Section2
A famous result of Kasteleyn [8] and Temperley and Fisher [18] counts the
number of domino tilings of a chessboard (or any other rectangular region).
In this section we explain Kasteleyn's proof.

「ドミノによるタイル張り」(京大･理)　36p.
http://www.ms.u-tokyo.ac.jp/~kazushi/proceedings/domino.pdf

「長方形領域のドミノタイル張りについて」(青学大･理工)　17p.
http://www.gem.aoyama.ac.jp/~kyo/sotsuken/2010/fujino_sotsuron_2010.pdf
40 ：: ■ドミノタイリング
https://ja.wikipedia.org/wiki/%E3%83%89%E3%83%9F%E3%83%8E%E3%82%BF%E3%82%A4%E3%83%AA%E3%83%B3%E3%82%B0

■Die Kasteleyn-Fisher-Temperley-Formel fur die Anzahl der Domino ...
http://oops.uni-oldenburg.de/1773/1/Bachelorarbeit%20von%20Ina%20Lammers.pdf
41 ：: ■平面充填（へいめんじゅうてん）

平面内を有限種類の平面図形（タイル）で隙間なく敷き詰める操作である
敷き詰めたタイルからなる平面全体を平面充填形という

平面敷き詰め、タイル貼り、タイリング (tiling) 、テセレーション
(tessellation) ともいう
ただし「平面」を明言しない場合は、曲面充填や、
場合によっては2次元以外の空間の充填を含む
広義のテセレーション等については、空間充填を参照
平面充填は広義の空間充填の一種で、2次元ユークリッド空間の
充填である

多面体は多角形による球面充填（曲面充填の一種）と
考えることができる
そのため、多角形による平面充填は多面体と共通点が多く、
便宜上多面体に含めて論じられることもある
42 ：: ルジンの問題（Luzin - のもんだい）とは、
正方形に関してニコライ・ルジン (Nikolai Luzin) が考えた問題である

「任意の正方形を、2個以上の全て異なる大きさの正方形に分割できるか」
という問題であり、ルジンはこの問題の解は存在しないと予想したが、
その後幾つかの例が発見された

2, 4, 6, 7, 8, 9, 11, 15, 16, 17, 18, 19, 24, 25, 27, 29, 33, 35, 37, 42, 50 の
計21枚の正方形
43 ：: Table[C(0,n-2 mod18)+3C(0,n-4)+3C(1,n-7)+7C(0,n-11)+C(1,n-16)+C(1,n-18),{n,1,27}]

{0, 1, 0, 3, 0, 0, 3, 3, 0, 0, 7, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0}

こういう数列を簡単に作る方法は？
44 ：: Table[{1-n(n-1)(n-2)(n-3)(n-4)(n-5)(n-6)(n-7)(n-8)(n-9)(n-10)(n-11)(n-12)/13!}/4,{n,0,13}]

Table[(1-C(0,n-13))/4,{n,0,13}]

同じ出力で遥かに式を短くできる
45 ：: 56を2進法表記で桁をリストアップし，
リスト長が8になるようにリストの左側にゼロを足し加える：

In[3]:=IntegerDigits[56, 2, 8]

Out[3]={0,0,1,1,1,0,0,0}
46 ：: FromDigits[{1,0,1,0,0,1,0,0,0}, 2]

328

Table[2n-1,{n,1,9}]+IntegerDigits[328, 2, 9]

{2, 3, 6, 7, 9, 12, 13, 15, 17}
47 ：: 3を法としたときの剰余：

Mod[{1, 2, 3, 4, 5, 6, 7}, 3]

{1,2,0,1,2,0,1}
48 ：: 2進値リストからもとの数を再生する：

IntegerDigits[56, 2, 8];

FromDigits[%, 2]
49 ：: a_n=1/4((-1)^n-(1+2i)(-i)^n-(1-2i)i^n+9)

1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2, 1, 3, 3, 2,
50 ：: 『与えられた数より小さい素数の個数について』

Chu-Vandermonde identity
51 ：: n個のものからk個取り出す場合の数と

k個取り残す場合の数は等しい
　　　　　　　　　　

C(n,k)=C(n,n-k)
52 ：: ■平方完成

y=ax^2-(-a+2)x-a-a+2
=a(x^2-(-a+2)x/a)-a-a+2
=a{(x-(-a+2)/(2a))^2-(-a+2)^2/(4a^2)}-a-a+2
=a(x-(-a+2)/(2a))^2-(-a+2)^2/(4a)-a-a+2
=a(x-(-a+2)/(2a))^2-(-a+2)^2/(4a)-2a+2
=a(x-(-a+2)/(2a))^2-(a^2-4a+4)/(4a)-2a+2
=a(x-(-a+2)/(2a))^2-(a^2-4a+4)/(4a)-(8a^2)/(4a)+(8a)/(4a)
=a(x-(-a+2)/(2a))^2-(a^2-4a+4+8a^2-8a)/(4a)
=a(x-(-a+2)/(2a))^2-(9a^2-12a+4)/(4a)
53 ：: トランプの束がある
2～10までの数字が描かれたカードが各スートに1枚ずつと、
ジョーカーのカードが24枚ある
全てを混ぜて無作為に切り直して12枚のカードを無作為に引いたとき
その12枚のカードのうちジョーカー以外にいずれも違う数字が
書かれている確率はいくらか

Sum[choose(24,k)*choose(9,12-k)*4^(12-k),{k,3,12}]/(choose(60,12))

Sum[C(24,k)C(9,12-k)4^(12-k),{k,3,12}]/(C(60,12))

出力　7371811052/66636135475
54 ：: FromDigits[{1,0,1,0,0,1,0,0}, 2]

164
55 ：: ガンマ関数とベータ関数
https://lecture.ecc.u-tokyo.ac.jp/~nkiyono/2006/miya-gamma.pdf
56 ：: 第一種の合流型超幾何関数（クンマー）

1Ｆ1[a; b; z] = 1+Σ[k=1, ∞] {a(a+1)････(a+k-1)/b(b+1)････(b+k-1)} z^k/k!

1Ｆ1[-n; -2n; z] = {n!/(2n)!} Σ[k=0, n] {(2n-k)!/(n-k)!k!} z^k
57 ：: ブリストル大学の数学者Andrew Booker氏が、
33を3つの立方数の合計で表すこと、すなわち
33＝x^3＋y^3＋z^3という方程式の解を求めることに成功した

(8866128975287528)^3+(-8778405442862239)^3+(-2736111468807040)^3=33

https://fabcross.jp/news/2019/20190507_33.html
58 ：: Table[choose(17,k-1)+choose(15,k-1)+choose(13,k-1)+choose(11,k-1)+choose(10,k-1)+choose(8,k-1)+choose(5,k-1)+choose(4,k-1)+choose(1,k-1),{k,1,20}]

chooseを一つにした式に変形できますか？

三つならできた

短軸有利☆

Table[sum[C(2n-1+C(0,n-2)+C(1,n-4),k-1),{n,1,9}],{k,1,20}]
59 ：: Table[C(0,n-2 mod4),{n,1,10}]

{0, 1, 0, 0, 0, 1, 0, 0, 0, 1}
60 ：: 長軸有利☆

Table[C(9,k-1)+C(7,k-1)+C(6,k-1)+C(3,k-1)+C(2,k-1),{k,1,12}]
Table[sum[C(2n-1+C(0,n-1)+C(0,n-3),k-1),{n,1,5}],{k,1,12}]
Table[sum[C(2n-1+C(0,3mod n),k-1),{n,1,5}],{k,1,12}]

{5, 27, 76, 140, 176, 153, 92, 37, 9, 1, 0, 0}

同じ出力で式が短くなってゆく
61 ：: Table[C(0,2mod n),{n,1,10}]

{1, 1, 0, 0, 0, 0, 0, 0, 0, 0}

Table[C(0,3mod n),{n,1,10}]

{1, 0, 1, 0, 0, 0, 0, 0, 0, 0}

Table[C(0,4mod n),{n,1,10}]

{1, 1, 0, 1, 0, 0, 0, 0, 0, 0}
62 ：: Table[C(0,5mod n),{n,1,10}]

{1, 0, 0, 0, 1, 0, 0, 0, 0, 0}

Table[C(0,6mod n),{n,1,10}]

{1, 1, 1, 0, 0, 1, 0, 0, 0, 0}

Table[C(0,7mod n),{n,1,10}]

{1, 0, 0, 0, 0, 0, 1, 0, 0, 0}
63 ：: Table[C(0,8mod n),{n,1,10}]

{1, 1, 0, 1, 0, 0, 0, 1, 0, 0}

Table[C(0,9mod n),{n,1,10}]

{1, 0, 1, 0, 0, 0, 0, 0, 1, 0}
64 ：: 【即時】金券五百円分とすかいらーく券を即ゲット　　　
https://pbs.twimg.com/media/D9F0S6KUcAAk_1s.jpg　
　　　　　
1. スマホでたいむばんくを入手 iOS https://t.co/ik17bynKNT Android https://t.co/uxTzFEk2ee　　　　　
2. 会員登録を済ませる　　　　
3. マイページへ移動する　　　　
4. 紹介コード → 入力する [Rirz Tu]（空白抜き）　　
　　
今なら更に16日23:59までの登録で倍額の600円を入手可
両方ゲットしてもおつりが来ます　　　　
　　　
　数分で終えられるのでぜひお試し下さい　　　　　　 👀
Rock54: Caution(BBR-MD5:b73a9cd27f0065c395082e3925dacf01)
65 ：: 短軸有利☆

Table[C(9,k-1)+C(7,k-1)+C(5,k-1)+C(4,k-1)+C(1,k-1),{k,1,12}]

Table[sum[C(2n-1+C(0,n-2),k-1),{n,1,5}],{k,1,12}]

{5, 26, 73, 133, 167, 148, 91, 37, 9, 1, 0, 0}

k-1を一つにして式を短縮
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sgssl;slg;ld;1
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sgssl;slg;ld;1
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sgssl;slg;ld;1
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sgssl;slg;ld;1
70 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
71 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
72 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
73 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
74 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
75 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;1
76 ：: >>64
ツイで見かけて既に貰ってる　　
77 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
78 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;1
79 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
80 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
81 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
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sgssl;slg;ld;1
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sgssl;slg;ld;1
84 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
85 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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sgssl;slg;ld;1
86 ：: 0qc3OXjXwBt1HD7Mt228BdYw7VinFEl43Zoc9tkhSu6hmEi1WEZ6OqB3FSe3k7L0qXrj8fNHrzBhhPTzD8WhAjrDXv1k55mQkf5uiOqjKuWUGgYFZuNZUagqAX9wNZzwaH4BlgoDtLscwycAwYQ7tuMa9CoGneWU5TXTTYEhxrUUJB0qmsR19UqgNcuwuN3oX8QyXNG
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UHzFXQXLWlU2JyYfOkcUjg40eeXu8e77qS4rgvjHWwxRbA1IKsNmn3CR9ePAuumjHnrWXeVuNStOENI4pHd7a7EwvoL8D7oenfbfeVoMT4Jho808YluyiYSEkfV5E6qPA5SiIWgl3G2zIc1NGsEW0nQuGTP504IXiRebMTKIEZyBdIqgQg1tOUYg7LzaYKiKixwj59ph
cILXg6ToA2TskeIlZraUTbzQ5PkR9lGnJjBFN7aSfv8vkCEpe9hYmrfF47H0RcNX5k3Y3i5xgHKhiNu5T8GeXfcYWpG6eLzIDAFy8DF39cqoofzCDnk8Ogt5q2H4cQNTgsDQDYYmFl2kKkYeX6CZ9LQruT8LprERVMyn0lUCYqfO4sQqWCu8kiVnpdZgd9QqdVcOT639
SDK4t8ql63OVBPRjJe2DvhC0BHjXErI2RCeGdMeBPD539aNqdFVIPGHN1NIVVDyTM4gfJbnFDgC5slKjSO17coT9jUfpOvezlHg9lXM95eftZiKzTx36T6C88TnssEI0tM3SKlDidfP9neTR3feD1cDGkYAzaPDCjyD4a2OcqNChan0XweFrq0xqQYc6Oi6am5DWfurQ
Kvr3idZa3OUlonClNyGyT3u2Xrtde47Cr6m4tG1j7AurlCjmUXvLPaQDQLlhymjaNIkWblKiKeVhk601XohEk7mNq3FyXjLGTJjx4csGI9MHt9vijbaaAQMFGIi8A28SQA1Ie3oELhvKeuLzK9ZYmGMEVqj4GtOgB719u1e1KHHqpfnGgwmMFMpRTjoTrEl4f9KFathh
kjasjfkajkljfklajjfjksdjksjalkjflkjasjfkjasjfkjajjs2333354994998989029929050295895028902802058299202095898582982092029209029029
sgssl;slg;ld;1
87 ：: ComplexExpand[(1+E^(I(1+n)Pi)+2n)/4]
88 ：: ■連続投稿・重複

連続投稿・コピー＆ペースト
連続投稿で利用者の会話を害しているものは削除対象になります
個々の内容に違いがあっても、荒らしを目的としていると判断したものは同様です
コピー＆ペーストやテンプレートの存在するものは、アレンジが施してあれば
残しますが、全く変更されていない・一部のみの変更で内容の変わらないもの、
スレッドの趣旨と違うもの、不快感を与えるのが目的なもの、
などは荒らしの意図があると判断して削除対象になります

※お手数ですが削除依頼できる方お願いします<(_ _)>
89 ：: ■DoS攻撃（ドスこうげき）（英：Denial of Service attack）

情報セキュリティにおける可用性を侵害する攻撃手法で、
ウェブサービスを稼働しているサーバやネットワークなどの
リソース（資源）に意図的に過剰な負荷をかけたり
脆弱性をついたりする事でサービスを妨害する攻撃、
サービス妨害攻撃である
90 ：: >>1は関係ないスレゴミを書き込むキチガイです。対応できるかたアクセス禁止をお願いします。
91 ：: https://rio2016.2ch.sc/test/read.cgi/math/1549700978/659-693
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/698-722
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/724-745
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/753-762
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/776-785
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/797-814
https://rio2016.2ch.sc/test/read.cgi/math/1560604951/24-33
https://rio2016.2ch.sc/test/read.cgi/math/1560604951/66-75
https://rio2016.2ch.sc/test/read.cgi/math/1560604951/77-86

数学板において連続大量レス投下によって

■初等関数研究所■
https://rio2016.2ch.sc/test/read.cgi/math/1549700978/

スレッドが機能不全に陥りました

他者の書き込みを妨害し、サーバー運営に多大な負荷をかけ
2ch掲示板において著しく公益性を損なう行為であるため
ここを巡回している運営の方々、厳重な対処をお願いします<(_ _)>
92 ：: 【ロビーのお約束】削除の要件（禁止されること）

荒らし依頼・ブラクラの張付け等第3者に迷惑がかかる行為
アダルト広告・勧誘・悪質な掲示板宣伝などのアドレス等張りつけ
煽り・煽りに対する返答・叩き・誹謗中傷等（差別発言等含む）
コピペ・アスキーアート等必要以上の張り付けまたは
第3者に迷惑が掛かる行為や発言であった場合は削除対象にします
93 ：: >>90
了解です。削除依頼出しておきます
94 ：: math：数学［スレッド削除］
https://qb5.2ch.sc/test/read.cgi/saku/1356355525/
95 ：: ■掲示板・スレッドの趣旨とは違う投稿

レス・発言

スレッドの趣旨から外れすぎ、議論または会話が成立しないほどの
状態になった場合は削除対象になります
故意にスレッドの運営・成長を妨害していると判断した場合も同様です

■投稿目的による削除対象

レス・発言

議論を妨げる煽り、不必要に差別の意図をもった発言、
第三者を不快にする暴言や排他的馴れ合い、
同一の内容を複数行書いたもの、
過度な性的妄想・下品である、等は削除対象とします
96 ：: >>93
おお、これは助かりまする
97 ：: 確率空間においては, A ∈ F を事象 (event) と呼ぶ.
98 ：: 100!中の二進数字の桁数を求める：

In[1]:=IntegerLength[100!, 2]

Out[1]=525
99 ：: 分数で表すと厳密な結果

小数で表すと近似になる
100 ：: ((-1)^n)(((-1)^n)n+n+4(-1)^n+2)/2

1 5 1 7 1 9 1 11 1 13 1 15 1 17 1

かなりエレガント☆

100～のスレッドの続きを読む

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