A378706 Array read by ascending antidiagonals: A(n,k) is the total semi-perimeter of n-Fibonacci polyominoes with k columns, where k > 0.
3, 4, 8, 5, 10, 16, 6, 12, 25, 33, 7, 14, 29, 54, 63, 8, 16, 33, 69, 118, 119, 9, 18, 37, 77, 152, 251, 219, 10, 20, 41, 85, 177, 335, 521, 398, 11, 22, 45, 93, 193, 390, 727, 1071, 714, 12, 24, 49, 101, 209, 433, 856, 1557, 2176, 1269, 13, 26, 53, 109, 225, 465, 948, 1859, 3297, 4380, 2237
Offset: 2
Examples
The array begins as: 3, 8, 16, 33, 63, 119, 219, 398, 714, 1269, ... 4, 10, 25, 54, 118, 251, 521, 1071, 2176, 4380, ... 5, 12, 29, 69, 152, 335, 727, 1557, 3297, 6931, ... 6, 14, 33, 77, 177, 390, 856, 1859, 4001, 8545, ... 7, 16, 37, 85, 193, 433, 948, 2065, 4463, 9581, ... ...
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 10.
Programs
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Mathematica
A[n_, k_]:=SeriesCoefficient[(n(1-y)y(1-2y-2y^n+3y^(n+1))-y(1-y^n)(-1+y-y^2+y^(n+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] (n*(1 - y)*y*(1 - 2*y - 2*y^n +3*y^(n+1)) - y*(1 - y^n)*(-1 + y - y^2 + y^(n+2)))/((1 - y)*(1 - 2*y + y^(n+1))^2).