cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A082297 Main diagonal of array A083861.

Original entry on oeis.org

0, 1, 5, 22, 105, 551, 3125, 18901, 120785, 809956, 5669525, 41253125, 310968905, 2421594449, 19434554165, 160416509794, 1359425390625, 11809173506531, 105013666399445, 954781355707373, 8865807366284825, 83996265859375000, 811222567266570805, 7980019803318567233
Offset: 0

Views

Author

Paul Barry, May 06 2003

Keywords

Crossrefs

Cf. A083859, A083861, A083862.

Programs

  • Maple
    T := proc(n, k) local v; option remember; if 0 <= n and k = 0 then v := 0; end if; if 0 <= n and k = 1 then v := 1; end if; if 0 <= n and 2 <= k then v := 5*T(n, k - 1) + (n - 6)*T(n, k - 2); end if; v; end proc;
seq(T(n, n), n = 0..40); # Petros Hadjicostas, Dec 25 2019

Formula

a(n) = (((5 + sqrt(4*n + 1))/2)^n - ((5 - sqrt(4*n + 1))/2)^n)/sqrt(4*n + 1).

A083856 Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 3, 1, 0, 1, 1, 4, 5, 5, 1, 0, 1, 1, 5, 7, 11, 8, 1, 0, 1, 1, 6, 9, 19, 21, 13, 1, 0, 1, 1, 7, 11, 29, 40, 43, 21, 1, 0, 1, 1, 8, 13, 41, 65, 97, 85, 34, 1, 0, 1, 1, 9, 15, 55, 96, 181, 217, 171, 55, 1
Offset: 0

Views

Author

Paul Barry, May 06 2003

Keywords

Comments

Row n >= 0 of the array gives the solution to the recurrence b(k) = b(k-1) + n*b(k-2) for k >= 2 with b(0) = 0 and b(1) = 1.

Examples

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1, 1,  1,  1,   1,   1,    1,    1,     1, ... [A057427]
  0, 1, 1,  2,  3,   5,   8,   13,   21,    34, ... [A000045]
  0, 1, 1,  3,  5,  11,  21,   43,   85,   171, ... [A001045]
  0, 1, 1,  4,  7,  19,  40,   97,  217,   508, ... [A006130]
  0, 1, 1,  5,  9,  29,  65,  181,  441,  1165, ... [A006131]
  0, 1, 1,  6, 11,  41,  96,  301,  781,  2286, ... [A015440]
  0, 1, 1,  7, 13,  55, 133,  463, 1261,  4039, ... [A015441]
  0, 1, 1,  8, 15,  71, 176,  673, 1905,  6616, ... [A015442]
  0, 1, 1,  9, 17,  89, 225,  937, 2737, 10233, ... [A015443]
  0, 1, 1, 10, 19, 109, 280, 1261, 3781, 15130, ... [A015445]
  ...

Links

Crossrefs

Rows include A057427 (n=0), A000045 (n=1), A001045 (n=2), A006130 (n=3), A006131 (n=4), A015440 (n=5), A015441 (n=6), A015442 (n=7), A015443 (n=8), A015445 (n=9).
Columns include A000012 (k=1,2), A000027 (k=3), A005408 (k=4), A028387 (k=5), A000567 (k=6), A106734 (k=7).
Cf. A083857 (binomial transform), A083859 (main diagonal), A083860 (first subdiagonal), A083861 (second binomial transform), A110112, A110113 (diagonal sums), A193376 (transposed variant), A172237 (transposed variant).

Programs

  • Julia
    function generalized_fibonacci(r, n)
   F = BigInt[1 r; 1 0]
   Fn = F^n
   Fn[2, 1]
end
for r in 0:6 println([generalized_fibonacci(r, n) for n in 0:9]) end # Peter Luschny, Mar 06 2017
  • Maple
    A083856_row := proc(r, n) local R; R := proc(n) option remember;
if n<=1 then n else R(n-1)+r*R(n-2) fi end: R(n) end:
for r from 0 to 9 do seq(A083856_row(r, n), n=0..9) od; # Peter Luschny, Mar 06 2017
  • Mathematica
    T[, 0] = 0; T[, 1|2] = 1; T[n_, k_] := T[n, k] = T[n, k-1] + n T[n, k-2];
Table[T[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 22 2018 *)

Formula

T(n, k) = (((1 + sqrt(4*n + 1))/2)^k - ((1 - sqrt(4*n + 1))/2)^k)/sqrt(4*n + 1). [corrected by Michel Marcus, Jun 25 2018]
From Thomas Baruchel, Jun 25 2018: (Start)
The g.f. for row n >= 0 is x/(1 - x - n*x^2).
The g.f. for column k >= 1 is g(k,x) = 1/(1-x) + Sum_{m = 1..floor((k-1)/2)} (1 - x)^(-1 - m) * binomial(k - 1 - m, m) * Sum_{i = 0..m} x^i * Sum_{j = 0..i} (-1)^j * (i - j)^m * binomial(1 + m, j).
The g.f. for column k >= 1 is also g(k,x) = 1 + Sum_{m = 1..floor((k+1)/2)} ((1 - x)^(-m) * binomial(k-m, m-1) * Sum_{j = 0..m} (-1)^j * binomial(m, j) * x^m * Phi(x, -m+1, -j+m)) + Sum_{s = 0..floor((k-1)/2)} binomial(k-s-1, s) * PolyLog(-s, x), where Phi is the Lerch transcendent function. (End)
T(n,k) = Sum_{i = 0..k} (-1)^(k+i) * binomial(k,i) * A083857(n,i). - Petros Hadjicostas, Dec 24 2019

Extensions

Various sections edited by Petros Hadjicostas, Dec 24 2019

A083857 Square array T(n,k) of binomial transforms of generalized Fibonacci numbers, read by ascending antidiagonals, with n, k >= 0.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 1, 3, 7, 0, 1, 3, 8, 15, 0, 1, 3, 9, 21, 31, 0, 1, 3, 10, 27, 55, 63, 0, 1, 3, 11, 33, 81, 144, 127, 0, 1, 3, 12, 39, 109, 243, 377, 255, 0, 1, 3, 13, 45, 139, 360, 729, 987, 511, 0, 1, 3, 14, 51, 171, 495, 1189, 2187, 2584, 1023, 0, 1, 3, 15, 57, 205, 648
Offset: 0

Views

Author

Paul Barry, May 06 2003

Keywords

Comments

Row n >= 0 of the array gives the solution to the recurrence b(k) = 3*b(k-1) + (n-2) * a(k-2) for k >= 2 with a(0) = 0 and a(1) = 1. These are the binomial transforms of the rows of the generalized Fibonacci numbers A083856.

Examples

			Array T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
  0, 1, 3,  7, 15,  31,  63,  127,  255, ...
  0, 1, 3,  8, 21,  55, 144,  377,  987, ...
  0, 1, 3,  9, 27,  81, 243,  729, 2187, ...
  0, 1, 3, 10, 33, 109, 360, 1189, 3927, ...
  0, 1, 3, 11, 39, 139, 495, 1763, 6279, ...
  0, 1, 3, 12, 45, 171, 648, 2457, 9315, ...
  ...

Links

Crossrefs

Rows include A000225 (n=0), A001906 (n=1), A000244 (n=2), A006190 (n=3), A007482 (n=4), A030195 (n=5), A015521 (n=6).
Cf. A083856, A083861 (binomial transform), A083862 (main diagonal).

Formula

T(n, k) = ((3 + sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) - ((3 - sqrt(4*n + 1))/2)^k / sqrt(4*n + 1) for n, k >= 0.
O.g.f. of row n >= 0: -x/(-1 + 3*x + (n-2)*x^2) . - R. J. Mathar, Nov 23 2007
T(n,k) = Sum_{i = 0..k} binomial(k,i)*A083856(n,i). - Petros Hadjicostas, Dec 24 2019

Extensions

Various sections edited by Petros Hadjicostas, Dec 24 2019
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.