A196265 Number of standard puzzles of shape 2 X n with support CK (see reference for precise definition).

1, 2, 4, 8, 26, 66, 276, 816, 4050, 13410, 75780, 274680, 1723050, 6735330, 46104660, 192296160, 1418802210, 6264006210, 49355252100, 229233450600, 1914861598650, 9309854203650, 81969299111700, 415483465597200, 3837397323409650, 20209910950879650

Offset

1,2

Comments

The Han reference contains many sequences not yet in the OEIS (as well as over 100 that are). This is the first one that was not already in the OEIS.

The sequence appears on pages 4 and 13 of the Han reference. a(1)=1 by convention. - Michael Somos, Jan 16 2014

Links

Alois P. Heinz, Table of n, a(n) for n = 1..500

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

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

Formula

a(n) = A235136(n-1) + A235136(n-2) if n > 1. - Michael Somos, Jan 16 2014

E.g.f. A(x) =: y satisfies 0 = -(1 + x)^2 + y * x - y' * (1 + 2*x + 2*x^2) + y'' * (1 + x) = (1 - x) + y' * (1 - x) - y'' * (1 + 2*x^2) + y''' * x. - Michael Somos, Jan 16 2014

Example

G.f. = x + 2*x^2 + 4*x^3 + 8*x^4 + 26*x^5 + 66*x^6 + 276*x^7 + 816*x^8 + ...

Mathematica

a[ n_] := If[ n < 2, Boole[n == 1], With[{m = Mod[n, 2]}, 2^(n - m) (Pochhammer[ 1/4 + m/2, (n - m)/2] - (-1)^ m Pochhammer[ -1/4 + m/2, (n - m)/2]) ]]; (* Michael Somos, Jan 16 2014 *)

Prog

(PARI) {a(n) = my(v=[1, 1]); if( n<2, n==1, for(k=1, n-1, v = [v[2], v[1] * (2*k-1)]); v[1] + v[2])}; /* Michael Somos, Jan 16 2014 */

Crossrefs

Cf. A235136.

Sequence in context: A261489 A286936 A323948 * A112285 A037170 A212409

Adjacent sequences: A196262 A196263 A196264 * A196266 A196267 A196268

Keyword

nonn

Author

N. J. A. Sloane, Oct 27 2011

Status

approved