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 A128619 #11 Mar 22 2024 17:42:04
%S A128619 1,0,1,2,0,2,0,3,0,3,5,0,5,0,5,0,8,0,8,0,8,13,0,13,0,13,0,13,0,21,0,
%T A128619 21,0,21,0,21,34,0,34,0,34,0,34,0,34,0,55,0,55,0,55,0,55,0,55
%N A128619 Triangle T(n, k) = A127647(n,k) * A128174(n,k), read by rows.
%C A128619 This triangle is different from A128618, which is equal to A128174 * A127647.
%H A128619 G. C. Greubel, <a href="/A128619/b128619.txt">Rows n = 1..100 of the triangle, flattened</a>
%F A128619 T(n, k) = A127647 * A128174, an infinite lower triangular matrix. In odd rows, n terms of F(n), 0, F(n),...; in the n-th row. In even rows, n terms of 0, F(n), 0,...; in the n-th row.
%F A128619 Sum_{k=1..n} T(n, k) = A128620(n-1).
%F A128619 From _G. C. Greubel_, Mar 16 2024: (Start)
%F A128619 T(n, k) = Fibonacci(n)*(1 + (-1)^(n+k))/2.
%F A128619 Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^n*A128620(n-1).
%F A128619 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1-(-1)^n)*A096140(floor((n + 1)/2)).
%F A128619 Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1 - (-1)^n)*( Fibonacci(n-1) + (-1)^floor((n-1)/2) * Fibonacci(floor((n-3)/2)) ). (End)
%e A128619 First few rows of the triangle are:
%e A128619 1;
%e A128619 0, 1;
%e A128619 2, 0, 2;
%e A128619 0, 3, 0, 3;
%e A128619 5, 0, 5, 0, 5;
%e A128619 0, 8, 0, 8, 0, 8;
%e A128619 13, 0, 13, 0, 13, 0, 13;
%e A128619 0, 21, 0, 21, 0, 21, 0, 21,
%e A128619 ...
%t A128619 Table[Fibonacci[n]*Mod[n+k+1,2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 16 2024 *)
%o A128619 (Magma) [((n+k+1) mod 2)*Fibonacci(n): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 17 2024
%o A128619 (SageMath) flatten([[((n+k+1)%2)*fibonacci(n) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Mar 17 2024
%Y A128619 Cf. A000045, A096140, A127646, A128174, A128610, A128618, A128620.
%Y A128619 Cf. A128620 (row sums).
%K A128619 nonn,tabl
%O A128619 1,4
%A A128619 _Gary W. Adamson_, Mar 14 2007