A128618 Triangle read by rows: A128174 * A127647 as infinite lower triangular matrices.

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, 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, 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

Offset: 1

Author

Gary W. Adamson, Mar 14 2007

Keywords

• nonn
• tabl

Comments

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

Examples

			First few rows of the triangle are:
  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,  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, 89;
  ...
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened

Crossrefs

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

Programs

Magma

[((n+k+1) mod 2)*Fibonacci(k): k in [1..n], n in [1..15]]; // G. C. Greubel, Mar 17 2024

Mathematica

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

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

Formula

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

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

From G. C. Greubel, Mar 17 2024: (Start)

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

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

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

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

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

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)

Extensions

a(6) corrected and more terms from Georg Fischer, May 30 2023