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.
%I A109754 #21 May 28 2022 12:47:14
%S A109754 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,
%T A109754 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,
%U A109754 28,37,47,55,55,0,1,10,10,17,23,33,45,60,76,89,89
%N A109754 Matrix defined by: a(i,0) = 0, a(i,j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0; read by ascending antidiagonals.
%C A109754 Lower triangular version is at A117501. - _Ross La Haye_, Apr 12 2006
%H A109754 G. C. Greubel, <a href="/A109754/b109754.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A109754 a(i, 0) = 0, a(i, j) = i*Fibonacci(j-1) + Fibonacci(j), for j > 0.
%F A109754 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.
%F A109754 G.f.: (x*(1 + ix))/(1 - x - x^2).
%F A109754 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
%e A109754 Table starts:
%e A109754 [0] 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
%e A109754 [1] 0, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...
%e A109754 [2] 0, 1, 3, 4, 7, 11, 18, 29, 47, 76, ...
%e A109754 [3] 0, 1, 4, 5, 9, 14, 23, 37, 60, 97, ...
%e A109754 [4] 0, 1, 5, 6, 11, 17, 28, 45, 73, 118, ...
%e A109754 [5] 0, 1, 6, 7, 13, 20, 33, 53, 86, 139, ...
%e A109754 [6] 0, 1, 7, 8, 15, 23, 38, 61, 99, 160, ...
%e A109754 [7] 0, 1, 8, 9, 17, 26, 43, 69, 112, 181, ...
%e A109754 [8] 0, 1, 9, 10, 19, 29, 48, 77, 125, 202, ...
%e A109754 [9] 0, 1, 10, 11, 21, 32, 53, 85, 138, 223, ...
%p A109754 A := (n, k) -> ifelse(k = 0, 0,
%p A109754 n*combinat:-fibonacci(k-1) + combinat:-fibonacci(k)):
%p A109754 seq(seq(A(n - k, k), k = 0..n), n = 0..6); # _Peter Luschny_, May 28 2022
%t A109754 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 *)
%Y A109754 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).
%Y A109754 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.
%Y A109754 a(2^i-2, j+1) = A118654(i, j), for i > 0.
%Y A109754 Cf. A117501.
%K A109754 nonn,tabl
%O A109754 0,9
%A A109754 _Ross La Haye_, Aug 11 2005; corrected Apr 14 2006
%E A109754 More terms from _G. C. Greubel_, Jan 07 2017