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MatsGranvik

n mod k recurrences 1 0

MatsGranvik
Aug 10th, 2014 (edited)
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  1. MatrixForm[Table[Table[If[n == k, 1, 0], {k, 1, 12}], {n, 1, 12}]]
  2.  
  3. MatrixForm[Table[Table[If[k == 1, 1, 0], {k, 1, 12}], {n, 1, 12}]]
  4.  
  5. Clear[t]
  6. t[1, 1] = 1;
  7. t[n_, k_] :=
  8. t[n, k] = If[k == 1, t[n, 2] + 1, If[n >= k, t[n - k + 1, 1], 0], 0]
  9. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  10.  
  11. Clear[t]
  12. t[1, 1] = 1;
  13. t[n_, k_] :=
  14. t[n, k] =
  15. If[k == 1, t[n, 2] + t[n, 3], If[n >= k, t[n - k + 1, 1], 0], 0]
  16. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  17.  
  18. Clear[t]
  19. t[1, 1] = 1;
  20. t[n_, k_] :=
  21. t[n, k] =
  22. If[k == 1, Sum[t[n, i], {i, 2, n}], If[n >= k, t[n - k + 1, 1], 0],
  23. 0]
  24. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  25.  
  26. Clear[t]
  27. t[1, 1] = 1;
  28. t[n_, k_] :=
  29. t[n, k] =
  30. If[k == 1, 2*t[n, 2]*If[n > 2, t[n, 3], 1],
  31. If[n >= k, t[n - k + 1, 1], 0], 0]
  32. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  33. Flatten[Table[Table[t[n, k], {k, 1, 1}], {n, 1, 12}]]
  34.  
  35. Clear[t]
  36. t[n_, 1] = 1;
  37. t[n_, k_] :=
  38. t[n, k] =
  39. If[n >= k,
  40. Sum[t[n - i, k - 1], {i, 1, n - 1}] -
  41. Sum[t[n - i, k], {i, 1, n - 1}], 0]
  42. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  43. (*TableForm[Table[Table[If[n>=k,Graphics[Disk[]],""],{k,1,12}],{n,1,\
  44. 12}]]*)
  45.  
  46. (*MatrixForm[Table[Table[If[Mod[n,k]==0,Graphics[Disk[]],""],{k,1,12}]\
  47. ,{n,1,12}]]*)
  48. MatrixForm[
  49. Table[Table[If[Mod[n, k] == 0, 1, 0], {k, 1, 12}], {n, 1, 12}]]
  50.  
  51. Clear[t]
  52. t[n_, 1] = 1;
  53. t[n_, k_] :=
  54. t[n, k] =
  55. If[n >= k,
  56. Sum[t[n - i, k - 1], {i, 1, k - 1}] -
  57. Sum[t[n - i, k], {i, 1, k - 1}], 0]
  58. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  59.  
  60. Clear[t]
  61. t[n_, 1] = 1;
  62. t[n_, k_] :=
  63. t[n, k] =
  64. If[n >= k,
  65. Sum[t[n - i, k - 1], {i, 1, k - 1}] -
  66. Sum[t[n - i, k], {i, 1, k - 1}], 0]
  67. MatrixForm[Inverse[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]]
  68.  
  69. Clear[t]
  70. t[1, 1] = 1;
  71. t[n_, k_] :=
  72. t[n, k] =
  73. If[k == 1, -Sum[t[n, k + i], {i, 1, n - 1}],
  74. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  75. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  76.  
  77.  
  78. Clear[t]
  79. t[1, 1] = 1;
  80. t[n_, k_] :=
  81. t[n, k] =
  82. If[k == 1, n/Product[t[n, k + i], {i, 1, n - 1}],
  83. If[Mod[n, k] == 0, t[n/k, 1], 1], 1]
  84. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  85.  
  86. Clear[t]
  87. t[1, 1] = 1;
  88. t[n_, k_] :=
  89. t[n, k] =
  90. If[k == 1, n - Sum[t[n, k + i], {i, 1, n - 1}],
  91. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  92. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  93.  
  94. Clear[t]
  95. t[1, 1] = 1;
  96. t[n_, k_] :=
  97. t[n, k] =
  98. If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}],
  99. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  100. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  101.  
  102. Clear[t]
  103. t[1, 1] = 1;
  104. t[n_, k_] :=
  105. t[n, k] =
  106. If[k == 1, 1/n - Sum[t[n, k + i], {i, 1, n - 1}],
  107. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  108. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  109.  
  110. Clear[t]
  111. t[1, 1] = 1;
  112. t[n_, k_] :=
  113. t[n, k] =
  114. If[k == 1, Sum[t[n, k + i], {i, 1, 2 - 1}],
  115. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  116. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  117.  
  118.  
  119. Clear[t]
  120. t[1, 1] = 1;
  121. t[n_, k_] :=
  122. t[n, k] =
  123. If[k == 1, 2*Sum[t[n, k + i], {i, 1, 2 - 1}],
  124. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  125. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  126.  
  127. Clear[t]
  128. t[1, 1] = 1;
  129. t[n_, k_] :=
  130. t[n, k] =
  131. If[k == 1, 2*Sum[t[n, k + i], {i, 1, 2 - 1}],
  132. If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  133. MatrixForm[Inverse[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]]
  134.  
  135.  
  136. Clear[t]
  137. t[1, 1] = 1;
  138. t[n_, k_] :=
  139. t[n, k] = If[k == 1, 3*t[n, 3], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  140. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  141.  
  142.  
  143. Clear[t]
  144. t[1, 1] = 1;
  145. t[n_, k_] :=
  146. t[n, k] = If[k == 1, 3*t[n, 3], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  147. MatrixForm[Inverse[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]]
  148.  
  149. Clear[t]
  150. t[1, 1] = 1;
  151. t[n_, k_] :=
  152. t[n, k] = If[k == 1, 2*t[n, 2], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]
  153. MatrixForm[Inverse[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]]
  154.  
  155. Clear[t]
  156. t[1, 1] = 1;
  157. t[n_, k_] :=
  158. t[n, k] =
  159. If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}],
  160. If[n >= k, t[n - k + 1, 1], 0], 0]
  161. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  162.  
  163. Clear[t]
  164. t[1, 1] = 1;
  165. t[n_, k_] :=
  166. t[n, k] =
  167. If[k == 1, -Sum[t[n, k + i], {i, 1, n - 1}],
  168. If[n >= k, t[n - k + 1, 1], 0], 0]
  169. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  170.  
  171. Clear[t]
  172. t[1, 1] = 1;
  173. t[n_, k_] :=
  174. t[n, k] = If[k == 1, t[n, 2], If[n >= k, t[n - k + 1, 1], 0], 0]
  175. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  176.  
  177. Clear[t]
  178. t[1, 1] = 1;
  179. t[n_, k_] :=
  180. t[n, k] = If[k == 1, t[n, 3], If[n >= k, t[n - k + 1, 1], 0], 0]
  181. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  182.  
  183.  
  184. Clear[t]
  185. t[n_, 1] = 1;
  186. t[n_, k_] :=
  187. t[n, k] =
  188. If[n >= k,
  189. Sum[t[n - i, k - 1], {i, 1, n - 1}] -
  190. Sum[t[n - i, k], {i, 1, n - 1}], 0]
  191. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  192.  
  193. MatrixForm[Table[Table[Mod[n, k], {k, 1, 12}], {n, 1, 12}]]
  194.  
  195. Clear[t]
  196. t[n_, 1] = 1;
  197. t[1, k_] = 1;
  198. t[n_, k_] :=
  199. t[n, k] =
  200. If[n >= k,
  201. Sum[t[n - i, k - 1], {i, 1, k - 1}] -
  202. Sum[t[n - i, k], {i, 1, k - 1}],
  203. Sum[t[k - i, n - 1], {i, 1, n - 1}] -
  204. Sum[t[k - i, n], {i, 1, n - 1}]]
  205. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  206.  
  207. Clear[t]
  208. t[n_, 1] = 1;
  209. t[1, k_] = 1;
  210. t[n_, k_] :=
  211. t[n, k] =
  212. If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[
  213. t[k - i, n], {i, 1, n - 1}]]
  214. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  215.  
  216. Clear[t]
  217. t[n_, 1] = 0;
  218. t[1, k_] = 1;
  219. t[n_, k_] := t[n, k] = If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], 1]
  220. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  221.  
  222. Clear[t]
  223. t[n_, 1] = 1;
  224. t[1, k_] = 1;
  225. t[n_, k_] :=
  226. t[n, k] =
  227. If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[
  228. t[k - i, n], {i, 1, n - 1}]]
  229. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  230. MatrixForm[Table[Sum[If[n >= k, t[n, k], 0], {k, 1, 12}], {n, 1, 12}]]
  231.  
  232. MatrixForm[Table[Table[n*k, {k, 1, 12}], {n, 1, 12}]]
  233.  
  234. MatrixForm[
  235. Table[Table[If[Mod[n, k] == 0, n, 0], {k, 1, 12}], {n, 1, 12}]]
  236.  
  237.  
  238. Clear[t]
  239. t[n_, 1] = 1;
  240. t[1, k_] = 1;
  241. t[n_, k_] :=
  242. t[n, k] =
  243. If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[
  244. t[k - i, n], {i, 1, n - 1}]]
  245. MatrixForm[Table[Table[t[n, k]/(n*k), {k, 1, 12}], {n, 1, 12}]]
  246. MatrixForm[
  247. Table[Sum[If[n >= k, t[n, k]/(n*k), 0], {k, 1, 12}]*n, {n, 1, 12}]]
  248. ListPlot[Table[
  249. Sum[If[n >= k, t[n, k]/(n*k), 0], {k, 1, 12}]*n, {n, 1, 12}],
  250. Filling -> Axis]
  251.  
  252.  
  253. Clear[t]
  254. t[n_, 1] = 1;
  255. t[1, k_] = 1;
  256. t[n_, k_] :=
  257. t[n, k] =
  258. If[n >= k, -Sum[t[n - i, k], {i, 1, k - 1}], -Sum[
  259. t[k - i, n], {i, 1, n - 1}]]
  260. ListPlot[Table[
  261. Sum[If[n >= k, t[n, k]/(n*k), 0], {k, 1, 12}]*n, {n, 1, 12}] -
  262. Table[Sum[If[n >= k, t[n, k], 0], {k, 1, 12}], {n, 1, 12}],
  263. Filling -> Axis]
  264.  
  265.  
  266. Clear[t]
  267. t[1, 1] = 1;
  268. t[n_, k_] :=
  269. t[n, k] =
  270. If[k == 1, n/Product[t[n, k + i], {i, 1, n - 1}],
  271. If[Mod[n, k] == 0, t[n/k, 1], 1], 1]
  272. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  273. MatrixForm[{Table[t[n, 1], {n, 1, 12}]}]
  274.  
  275. Clear[t]
  276. t[1, 1] = Log[1];
  277. t[n_, k_] :=
  278. t[n, k] =
  279. If[k == 1, Log[n] - Sum[t[n, k + i], {i, 1, n - 1}],
  280. If[Mod[n, k] == 0, t[n/k, 1], Log[1]], Log[1]]
  281. MatrixForm[Table[Table[FullSimplify[t[n, k]], {k, 1, 12}], {n, 1, 12}]]
  282. MatrixForm[{Table[FullSimplify[t[n, 1]], {n, 1, 12}]}]
  283.  
  284.  
  285. MatrixForm[
  286. Table[Table[
  287. If[Mod[n, k] == 0, n/k*MoebiusMu[n/k], 0], {k, 1, 12}], {n, 1,
  288. 12}]]
  289.  
  290.  
  291. Clear[t]
  292. t[n_, 1] = 1;
  293. t[n_, k_] :=
  294. t[n, k] =
  295. If[n >= k,
  296. Sum[t[n - i, k - 1], {i, 1, k - 1}] -
  297. 0*Sum[t[n - i, k], {i, 1, k - 1}], 0]
  298. MatrixForm[Table[Table[t[n, k], {k, 1, 12}], {n, 1, 12}]]
  299.  
  300. (*Recursive formula for the DivisorSigma function:*)(*Mathematica*)(*start*)
  301. Clear[t, s, n, k, z];
  302. z = 1;
  303. nn = 12;
  304. t[1, 1] = 1;
  305. t[n_, k_] :=
  306. t[n, k] =
  307. If[k == 1, 1 - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}],
  308. If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], 0]; MatrixForm[
  309. Table[Table[Limit[t[n, k], s -> z], {k, 1, nn}], {n, 1, nn}]] (*end*)
  310.  
  311.  
  312. (*Recursive formula for the DivisorSigma function Lagarias Riemann hypothesis:*)
  313. (*Mathematica*)
  314. (*start*)
  315. Clear[t, s, n, k, z];
  316. z = 1;
  317. nn = 142;
  318. t[1, 1] = 1;
  319. t[n_, k_] :=
  320. t[n, k] =
  321. If[k == 1, n - Sum[t[n, k + i]/(i + 1)^(s - 1), {i, 1, n - 1}],
  322. If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], 0]; MatrixForm[
  323. A = Table[Table[Limit[t[n, k], s -> z], {k, 1, nn}], {n, 1, nn}]];
  324. A[[All, 1]]
  325. (*https://oeis.org/A000203*)
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