A128618 - OEIS

%I #13 Mar 22 2024 17:41:51

%S 1,0,1,1,0,2,0,1,0,3,1,0,2,0,5,0,1,0,3,0,8,1,0,2,0,5,0,13,0,1,0,3,0,8,

%T 0,21,1,0,2,0,5,0,13,0,34,0,1,0,3,0,8,0,21,0,55,1,0,2,0,5,0,13,0,34,0,

%U 89,0,1,0,3,0,8,0,21,0,55,0,144,1,0,2,0,5,0,13,0,34,0,89,0,233

%N Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

%C This triangle is different from A128619, which is A128619 = A127647 * A128174.

%H G. C. Greubel, <a href="/A128618/b128618.txt">Rows n = 1..100 of the triangle, flattened</a>

%F By columns, Fibonacci(k) interspersed with alternate zeros in every column, k=1,2,3,...

%F Sum_{k=1..n} T(n, k) = A052952(n-1) (row sums).

%F From _G. C. Greubel_, Mar 17 2024: (Start)

%F T(n, k) = (1/2)*(1 + (-1)^(n+k))*Fibonacci(k).

%F T(n, n) = A000045(n).

%F T(2*n-1, n) = (1/2)*(1-(-1)^n)*A000045(n).

%F Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (-1)^(n-1)*A052952(n-1).

%F Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = (1/2)*(1 - (-1)^n)*(Fibonacci((n+ 5)/2) - 1).

%F Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1, k) = (1/2)*(1-(-1)^n) *  A355020(floor((n-1)/2)). (End)

%e First few rows of the triangle are:

%e   1;

%e   0, 1;

%e   1, 0, 2;

%e   0, 1, 0, 3;

%e   1, 0, 2, 0, 5;

%e   0, 1, 0, 3, 0, 8;

%e   1, 0, 2, 0, 5, 0, 13;

%e   0, 1, 0, 3, 0, 8,  0, 21;

%e   1, 0, 2, 0, 5, 0, 13,  0, 34;

%e   0, 1, 0, 3, 0, 8,  0, 21,  0, 55;

%e   1, 0, 2, 0, 5, 0, 13,  0, 34,  0, 89;

%e   ...

%t Table[Fibonacci[k]*Mod[n-k+1,2], {n,15}, {k,n}]//Flatten (* _G. C. Greubel_, Mar 17 2024 *)

%o (Magma) [((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // _G. C. Greubel_, Mar 17 2024

%o (SageMath) flatten([[((n-k+1)%2)*fibonacci(k) for k in range(1,n+1)] for n in range(1,16)]) # _G. C. Greubel_, Mar 17 2024

%Y Cf. A000045, A052952, A127647, A128174, A128619, A355020.

%K nonn,tabl

%O 1,6

%A _Gary W. Adamson_, Mar 14 2007

%E a(6) corrected and more terms from _Georg Fischer_, May 30 2023