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

2,5

Links

Table of n, a(n) for n=2..67.

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.

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.

Example

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, ...
  ...

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

Crossrefs

Cf. A023610, A378704, A378706, A378716.

Sequence in context: A072689 A021972 A193451 * A335809 A322215 A244492

Adjacent sequences: A378701 A378704 A378706 * A378708 A378709 A378712

Keyword

nonn,tabl,new

Author

Stefano Spezia, Dec 05 2024

Status

approved