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

2,1

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 10.

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).

Example

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, ...
  ...

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

Crossrefs

Cf. A378704, A378707, A378716.

Sequence in context: A050274 A262951 A288091 * A057926 A078766 A336840

Adjacent sequences: A378700 A378701 A378704 * A378707 A378708 A378709

Keyword

nonn,tabl,new

Author

Stefano Spezia, Dec 05 2024

Status

approved