API tools faq
paste
MatsGranvik

Mathematica RSolve applied to Divisibility recurrence

MatsGranvik
Aug 31st, 2021 (edited)
144
0
Never
Add comment
Not a member of Pastebin yet? Sign Up, it unlocks many cool features!
text 4.39 KB | None | 0 0
raw download clone embed print report
  1. (*start*)
  2. RSolve[{y[n] == 1, y[n - 1] - z[n - 1] == z[n], y[1] == 1,
  3. z[1] == 0}, {y, z}, n]
  4. ListPlot[Transpose @ Table[{y[n], 2*z[n]} /. First[%], {n, 0, 15}],
  5. Filling -> Axis]
  6.  
  7.  
  8.  
  9. Clear[a, b, c]
  10. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  11. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n], a[1] == 1,
  12. b[1] == 0, c[1] == 0, c[2] == 0}, {a, b, c}, n]
  13.  
  14.  
  15. Clear[a, b, c, d]
  16. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  17. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n],
  18. c[n - 3] + c[n - 2] + c[n - 1] - d[n - 3] - d[n - 2] - d[n - 1] ==
  19. d[n], a[1] == 1, b[1] == 0, c[1] == 0, c[2] == 0, d[1] == 0,
  20. d[2] == 0, d[3] == 0}, {a, b, c, d}, n]
  21.  
  22.  
  23. Clear[a, b, c, d]
  24. FullSimplify[
  25. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  26. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n],
  27. c[n - 3] + c[n - 2] + c[n - 1] - d[n - 3] - d[n - 2] - d[n - 1] ==
  28. d[n], a[1] == 1, b[1] == 0, c[1] == 0, c[2] == 0, d[1] == 0,
  29. d[2] == 0, d[3] == 0}, {a, b, c, d}, n]]
  30.  
  31. Clear[a, b, c, d, f]
  32. FullSimplify[
  33. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  34. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n],
  35. c[n - 3] + c[n - 2] + c[n - 1] - d[n - 3] - d[n - 2] - d[n - 1] ==
  36. d[n], d[n - 4] + d[n - 3] + d[n - 2] + d[n - 1] - f[n - 4] -
  37. f[n - 3] - f[n - 2] - f[n - 1] == f[n], a[1] == 1, b[1] == 0,
  38. c[1] == 0, c[2] == 0, d[1] == 0, d[2] == 0, d[3] == 0, f[1] == 0,
  39. f[2] == 0, f[3] == 0, f[4] == 0}, {a, b, c, d, f}, n]]
  40.  
  41. Clear[a, b, c, d, f, g]
  42. FullSimplify[
  43. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  44. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n],
  45. c[n - 3] + c[n - 2] + c[n - 1] - d[n - 3] - d[n - 2] - d[n - 1] ==
  46. d[n], d[n - 4] + d[n - 3] + d[n - 2] + d[n - 1] - f[n - 4] -
  47. f[n - 3] - f[n - 2] - f[n - 1] == f[n],
  48. f[n - 5] + f[n - 4] + f[n - 3] + f[n - 2] + f[n - 1] - g[n - 5] -
  49. g[n - 4] - g[n - 3] - g[n - 2] - g[n - 1] == g[n], a[1] == 1,
  50. b[1] == 0, c[1] == 0, c[2] == 0, d[1] == 0, d[2] == 0, d[3] == 0,
  51. f[1] == 0, f[2] == 0, f[3] == 0, f[4] == 0, g[1] == 0, g[2] == 0,
  52. g[3] == 0, g[4] == 0, g[5] == 0}, {a, b, c, d, f, g}, n]]
  53.  
  54.  
  55. Clear[a, b, c, d, f, g, h]
  56. FullSimplify[
  57. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  58. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n],
  59. c[n - 3] + c[n - 2] + c[n - 1] - d[n - 3] - d[n - 2] - d[n - 1] ==
  60. d[n], d[n - 4] + d[n - 3] + d[n - 2] + d[n - 1] - f[n - 4] -
  61. f[n - 3] - f[n - 2] - f[n - 1] == f[n],
  62. f[n - 5] + f[n - 4] + f[n - 3] + f[n - 2] + f[n - 1] - g[n - 5] -
  63. g[n - 4] - g[n - 3] - g[n - 2] - g[n - 1] == g[n],
  64. g[n - 6] + g[n - 5] + g[n - 4] + g[n - 3] + g[n - 2] + g[n - 1] -
  65. h[n - 6] - h[n - 5] - h[n - 4] - h[n - 3] - h[n - 2] -
  66. h[n - 1] == h[n], a[1] == 1, b[1] == 0, c[1] == 0, c[2] == 0,
  67. d[1] == 0, d[2] == 0, d[3] == 0, f[1] == 0, f[2] == 0, f[3] == 0,
  68. f[4] == 0, g[1] == 0, g[2] == 0, g[3] == 0, g[4] == 0, g[5] == 0,
  69. h[1] == 0, h[2] == 0, h[3] == 0, h[4] == 0, h[5] == 0,
  70. h[6] == 0}, {a, b, c, d, f, g, h}, n]]
  71. (*end*)
  72.  
  73.  
  74. nn = 20;
  75. Integrate[
  76. 1/nn*(1 + Sum[2*Cos[(2*k*n \[Pi])/nn], {k, 1, (nn - 1)/2}]), n]
  77. Plot[Re[%], {n, 1, 120}]
  78. 1/nn*(1 + Sum[2*Cos[(2*k*n \[Pi])/nn], {k, 1, (nn - 1)/2}])
  79. Plot[Re[%], {n, 1, 120}, PlotRange -> {-1, 2}]
  80.  
  81. Clear[n, nn];
  82. nn = 20;
  83. Integrate[
  84. Sum[2/nn*Exp[I*n*Pi/nn]*Cos[n*(2*k - 1)*Pi/nn], {k, 1, nn/2}], n]
  85. Plot[Re[%], {n, 1, 200}]
  86. Sum[2/nn*Exp[I*n*Pi/nn]*Cos[n*(2*k - 1)*Pi/nn], {k, 1, nn/2}]
  87. Plot[Re[%], {n, 1, 200}, PlotRange -> {-1, 2}]
  88.  
  89.  
  90. (*start*)
  91. Clear[a, b, c, d, n]
  92. FullSimplify[
  93. RSolve[{a[n] == 1, a[n - 1] - b[n - 1] == b[n],
  94. b[n - 2] + b[n - 1] - c[n - 2] - c[n - 1] == c[n],
  95. c[n - 3] + c[n - 2] + c[n - 1] - d[n - 3] - d[n - 2] - d[n - 1] ==
  96. d[n], a[1] == 1, b[1] == 0, c[1] == 0, c[2] == 0, d[1] == 0,
  97. d[2] == 0, d[3] == 0}, {a, b, c, d}, n]]
  98.  
  99. FullSimplify[
  100. DSolve[{a[n] == 1, a'[n] - b'[n] == b[n],
  101. b''[n] + b'[n] - c''[n] - c'[n] == c[n],
  102. c'''[n] + c''[n] + c'[n] - d'''[n] - d''[n] - d'[n] == d[n],
  103. a[1] == 1, b[1] == 0, c[1] == 0, c[2] == 0, d[1] == 0, d[2] == 0,
  104. d[3] == 0}, {a[n], b[n], c[n], d[n]}, n]]
  105.  
  106. FullSimplify[
  107. DSolve[{a[n] == n, a'[n] - b'[n] == b[n],
  108. b''[n] + b'[n] - c''[n] - c'[n] == c[n],
  109. c'''[n] + c''[n] + c'[n] - d'''[n] - d''[n] - d'[n] == d[n],
  110. a[1] == 1, b[1] == 0, c[1] == 0, c[2] == 0, d[1] == 0, d[2] == 0,
  111. d[3] == 0}, {a[n], b[n], c[n], d[n]}, n]]
  112. (*end*)
Add Comment
Please, Sign In to add comment
Public Pastes
We use cookies for various purposes including analytics. By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy.  OK, I Understand
Not a member of Pastebin yet?
Sign Up, it unlocks many cool features!