A378706 - OEIS

%I #11 Dec 06 2024 11:30:41

%S 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,

%T 152,251,219,10,20,41,85,177,335,521,398,11,22,45,93,193,390,727,1071,

%U 714,12,24,49,101,209,433,856,1557,2176,1269,13,26,53,109,225,465,948,1859,3297,4380,2237

%N Array read by ascending antidiagonals: A(n,k) is the total semi-perimeter of n-Fibonacci polyominoes with k columns, where k > 0.

%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 10.

%F 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).

%e The array begins as:

%e   3,  8, 16, 33,  63, 119, 219,  398,  714, 1269, ...

%e   4, 10, 25, 54, 118, 251, 521, 1071, 2176, 4380, ...

%e   5, 12, 29, 69, 152, 335, 727, 1557, 3297, 6931, ...

%e   6, 14, 33, 77, 177, 390, 856, 1859, 4001, 8545, ...

%e   7, 16, 37, 85, 193, 433, 948, 2065, 4463, 9581, ...

%e   ...

%t 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

%Y Cf. A378704, A378707, A378716.

%K nonn,tabl

%O 2,1

%A _Stefano Spezia_, Dec 05 2024