OFFSET
0,9
COMMENTS
Lower triangular version is at A117501. - Ross La Haye, Apr 12 2006
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
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
EXAMPLE
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, ...
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 *)
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.
KEYWORD
nonn,tabl
AUTHOR
Ross La Haye, Aug 11 2005; corrected Apr 14 2006
EXTENSIONS
More terms from G. C. Greubel, Jan 07 2017
STATUS
approved