A083856 Square array T(n,k) of generalized Fibonacci numbers, read by antidiagonals upwards (n, k >= 0).
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
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
- A. G. Shannon and J. V. Leyendekkers, The Golden Ratio family and the Binet equation, Notes on Number Theory and Discrete Mathematics, 21(2) (2015), 35-42.
Crossrefs
Programs
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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
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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
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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
Comments