A378716 Triangle read by rows: T(n,k) is the number of k-Fibonacci polyominoes with an area of n, with k > 1.

1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 0, 0, 1, 3, 0, 1, 0, 0, 1, 4, 2, 1, 0, 0, 0, 1, 5, 3, 1, 1, 0, 0, 0, 1, 7, 1, 1, 1, 0, 0, 0, 0, 1, 9, 5, 2, 0, 1, 0, 0, 0, 0, 1, 12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1, 16, 3, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 21, 10, 3, 3, 0, 1, 0, 0, 0, 0, 0, 0, 1

Offset: 3

Author

Stefano Spezia, Dec 05 2024

Keywords

• nonn
• tabl

Examples

			The triangle begins as:
   1;
   1, 1;
   1, 0, 1;
   2, 1, 0, 1;
   2, 2, 0, 0, 1;
   3, 0, 1, 0, 0, 1;
   4, 2, 1, 0, 0, 0, 1;
   5, 3, 1, 1, 0, 0, 0, 1;
   7, 1, 1, 1, 0, 0, 0, 0, 1;
   9, 5, 2, 0, 1, 0, 0, 0, 0, 1;
  12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1;
  ...
		

Links

• Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, Generating Trees and Fibonacci Polyominoes, arXiv:2411.17812 [math.CO], 2024. See page 9.

Crossrefs

Cf. A079957 (k=3), A182097 (k=2), A378704, A378706, A378707.

Programs

Mathematica

T[n_, k_]:=SeriesCoefficient[1/(1-Sum[x^((k+i)(k-i+1)/2), {i, k}]), {x, 0, n}]; Table[T[n, k], {n, 2, 14}, {k, 2,n}]//Flatten

Formula

T(n, k) = [x^n] 1/(1 - Sum_{i=1..k} x^((k+i)*(k-i+1)/2) ).