A378707 Array read by ascending antidiagonals: A(n,k) is the total number of inner points of n-Fibonacci polyominoes with k columns, where k > 0.
0, 0, 1, 0, 3, 3, 0, 5, 10, 7, 0, 7, 18, 26, 15, 0, 9, 26, 50, 63, 30, 0, 11, 34, 74, 124, 143, 58, 0, 13, 42, 98, 190, 296, 313, 109, 0, 15, 50, 122, 254, 457, 679, 668, 201, 0, 17, 58, 146, 318, 622, 1070, 1517, 1398, 365, 0, 19, 66, 170, 382, 782, 1461, 2439, 3325, 2883, 655
Offset: 2
Examples
The array begins as: 0, 1, 3, 7, 15, 30, 58, 109, 201, 365, ... 0, 3, 10, 26, 63, 143, 313, 668, 1398, 2883, ... 0, 5, 18, 50, 124, 296, 679, 1517, 3325, 7184, ... 0, 7, 26, 74, 190, 457, 1070, 2439, 5453, 12013, ... 0, 9, 34, 98, 254, 622, 1461, 3361, 7583, 16857, ... 0, 11, 42, 122, 318, 782, 1854, 4272, 9681, 21615, ... 0, 13, 50, 146, 382, 942, 2238, 5182, 11754, 26302, ... ...
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 12.
Programs
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Mathematica
A[n_, k_]:=SeriesCoefficient[y((6-4n)y-(2-4n)y^2-(3-n)n y^n-2(2-n)^2y^(n+1)+(2-5n+n^2)y^(n+2)+2y^(2n+1))/(2(-1+y)(1-2y+y^(n+1))^2), {y, 0, k}]; Table[A[n-k+1, k], {n, 2, 12}, {k, n-1}]//Flatten
Formula
A(n, k) = [y^k] y*((6 - 4*n)*y - (2 - 4*n)*y^2 - (3 - n)*n*y^n -2*(2 - n)^2*y^(n+1) + (2 - 5*n + n^2)*y^(n+2) + 2*y^(2n+1))/(2*(-1 + y)*(1 - 2*y + y^(n+1))^2).
A(2, n) = A023610(n-2) for n > 1.