A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.
0, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 3, 3, 0, 1, 4, 4, 5, 5, 0, 1, 5, 5, 7, 8, 8, 0, 1, 6, 6, 9, 11, 13, 13, 0, 1, 7, 7, 11, 14, 18, 21, 21, 0, 1, 8, 8, 13, 17, 23, 29, 34, 34, 0, 1, 9, 9, 15, 20, 28, 37, 47, 55, 55, 0, 1, 10, 10, 17, 23, 33, 45, 60, 76, 89, 89
Offset: 0
Examples
Table starts: [0] 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... [1] 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... [2] 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, ... [3] 0, 1, 4, 5, 9, 14, 23, 37, 60, 97, ... [4] 0, 1, 5, 6, 11, 17, 28, 45, 73, 118, ... [5] 0, 1, 6, 7, 13, 20, 33, 53, 86, 139, ... [6] 0, 1, 7, 8, 15, 23, 38, 61, 99, 160, ... [7] 0, 1, 8, 9, 17, 26, 43, 69, 112, 181, ... [8] 0, 1, 9, 10, 19, 29, 48, 77, 125, 202, ... [9] 0, 1, 10, 11, 21, 32, 53, 85, 138, 223, ...
Links
- G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Crossrefs
Rows: A000045(j); A000045(j+1), for j > 0; A000032(j), for j > 0; A000285(j-1), for j > 0; A022095(j-1), for j > 0; A022096(j-1), for j > 0; A022097(j-1), for j > 0. Diagonals: a(i, i) = A094588(i); a(i, i+1) = A007502(i+1); a(i, i+2) = A088209(i); Sum[a(i-j, j), {j=0...i}] = A104161(i). a(i, j) = A101220(i, 0, j).
Rows 7 - 19: A022098(j-1), for j > 0; A022099(j-1), for j > 0; A022100(j-1), for j > 0; A022101(j-1), for j > 0; A022102(j-1), for j > 0; A022103(j-1), for j > 0; A022104(j-1), for j > 0; A022106(j-1), for j > 0; A022107(j-1), for j > 0; A022108(j-1), for j > 0; A022109(j-1), for j > 0; A022110(j-1), for j > 0.
a(2^i-2, j+1) = A118654(i, j), for i > 0.
Cf. A117501.
Programs
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Maple
A := (n, k) -> ifelse(k = 0, 0, n*combinat:-fibonacci(k-1) + combinat:-fibonacci(k)): seq(seq(A(n - k, k), k = 0..n), n = 0..6); # Peter Luschny, May 28 2022 -
Mathematica
T[n_, 0]:= 0; T[n_, 1]:= 1; T[n_, 2]:= n - 1; T[n_, 3]:= n - 1; T[n_, n_]:= Fibonacci[n]; T[n_, k_]:= T[n, k] = T[n - 1, k - 1] + T[n - 2, k - 2]; Table[T[n, k], {n, 0, 15}, {k, 0, n}] (* G. C. Greubel, Jan 07 2017 *)
Formula
a(i, 0) = 0, a(i, j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0.
a(i, 0) = 0, a(i, 1) = 1, a(i, 2) = i+1, a(i, j) = a(i, j-1) + a(i, j-2), for j > 2.
G.f.: (x*(1 + ix))/(1 - x - x^2).
Sum_{j=0..i+1} a(i-j+1, j) - Sum_{j=0..i} a(i-j, j) = A001595(i). - Ross La Haye, Jun 03 2006
Extensions
More terms from G. C. Greubel, Jan 07 2017
Comments