A378704 Array read by ascending antidiagonals: A(n, k) is the total area of n-Fibonacci polyominoes with k columns, where k > 0.

2, 3, 7, 4, 11, 16, 5, 15, 31, 35, 6, 19, 43, 73, 70, 7, 23, 55, 111, 168, 136, 8, 27, 67, 143, 261, 370, 256, 9, 31, 79, 175, 351, 602, 790, 473, 10, 35, 91, 207, 431, 816, 1350, 1658, 860, 11, 39, 103, 239, 511, 1023, 1865, 2966, 3425, 1545, 12, 43, 115, 271, 591, 1215, 2346, 4178, 6414, 6989, 2748

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 pages 8-9.

Formula

A(n, k) = [y^k] y*(n^2*(1 - y)^2*y^n + 2*y*(1 - y^n) - n(1 - y)*(2- y^n + y^(n+1)))/(2*(-1 + y)*(1 - 2*y + y^(n+1))^2).

A(n, n-1) = A356888(n) - 1.

Example

The array begins as:
  2,  7, 16,  35,  70,  136,  256, ...
  3, 11, 31,  73, 168,  370,  790, ...
  4, 15, 43, 111, 261,  602, 1350, ...
  5, 19, 55, 143, 351,  816, 1865, ...
  6, 23, 67, 175, 431, 1023, 2346, ...
  7, 27, 79, 207, 511, 1215, 2815, ...
  ...

Mathematica

A[n_, k_]:=SeriesCoefficient[y(n^2(1-y)^2y^n+2y(1-y^n)-n(1-y)(2-y^n+y^(n+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. A004767 (k=2), A017605 (k=3), A356888, A378706, A378707, A378716.

Sequence in context: A357482 A026259 A168087 * A168085 A287950 A287628

Adjacent sequences: A378701 A378702 A378703 * A378705 A378706 A378707

Keyword

nonn,tabl

Author

Stefano Spezia, Dec 04 2024

Status

approved