A027989 a(n) = self-convolution of row n of array T given by A027926.

1, 3, 10, 33, 105, 324, 977, 2895, 8462, 24465, 70101, 199368, 563425, 1583643, 4430290, 12342849, 34262337, 94800780, 261545777, 719697255, 1975722326, 5412138033, 14796520365, 40380240528, 110016825025, 299285288499

Offset

0,2

Comments

a(n) is the number of all columns in stack polyominoes of perimeter 2n+4. - Emanuele Munarini, Apr 07 2011

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Guo-Niu Han, Enumeration of Standard Puzzles, 2011. [Cached copy]

Guo-Niu Han, Enumeration of Standard Puzzles, arXiv:2006.14070 [math.CO], 2020.

Index entries for linear recurrences with constant coefficients, signature (6,-11,6,-1).

Formula

a(n) = (2/5)*(n + 1)*F(2*n+3) + (1/5)*F(2*n+2) - (4/5)*(n + 1)*F(2*n), where F(n) = A000045(n). - Ralf Stephan, May 13 2004

From Emanuele Munarini, Apr 07 2011: (Start)

a(n) = ((4*n + 5)*F(2*n+1) - (2*n + 1)*F(2*n))/5, where F(n) = A000045(n).

a(n) = Sum_{k=0..n} binomial(2*n-k, k)*(k + 1).

G.f.: (1 - 3*x + 3*x^2)/(1 - 3*x + x^2)^2.

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) - a(n-4). (End)

Mathematica

Table[((5+4n)Fibonacci[1+2n]-(1+2n)Fibonacci[2n])/5, {n, 0, 20}] [Emanuele Munarini, Apr 07 2011]

Prog

(Maxima) makelist(((5+4*n)*fib(1+2*n)-(1+2*n)*fib(2*n))/5, n, 0, 20); /* Emanuele Munarini, Apr 07 2011 */
(PARI) Vec((1-3*x+3*x^2)/(1-3*x+x^2)^2+O(x^66)) /* Joerg Arndt, Apr 08 2011 */

Crossrefs

Cf. A027926, A054142, A172991, A188648.

Sequence in context: A062454 A121523 A115240 * A096483 A093043 A061566

Adjacent sequences: A027986 A027987 A027988 * A027990 A027991 A027992

Keyword

nonn

Author

Clark Kimberling

Status

approved