A378716 - OEIS

%I #10 Dec 06 2024 11:30:50

%S 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,

%T 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,

%U 3,2,0,0,1,0,0,0,0,0,1,21,10,3,3,0,1,0,0,0,0,0,0,1

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

%H Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, <a href="https://arxiv.org/abs/2411.17812">Generating Trees and Fibonacci Polyominoes</a>, arXiv:2411.17812 [math.CO], 2024. See page 9.

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

%e The triangle begins as:

%e    1;

%e    1, 1;

%e    1, 0, 1;

%e    2, 1, 0, 1;

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

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

%e    4, 2, 1, 0, 0, 0, 1;

%e    5, 3, 1, 1, 0, 0, 0, 1;

%e    7, 1, 1, 1, 0, 0, 0, 0, 1;

%e    9, 5, 2, 0, 1, 0, 0, 0, 0, 1;

%e   12, 5, 1, 1, 1, 0, 0, 0, 0, 0, 1;

%e   ...

%t 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

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

%K nonn,tabl,new

%O 3,7

%A _Stefano Spezia_, Dec 05 2024