A092439 Sequence arising from enumeration of domino tilings of Aztec Pillow-like regions.

0, 0, 6, 30, 140, 560, 2058, 7098, 23472, 75372, 237182, 735878, 2260596, 6896136, 20933778, 63325170, 191089112, 575626052, 1731858246, 5206059774, 15640198620, 46966732320, 140996664986, 423191320490, 1269993390720

Offset: 0

Author

Christopher Hanusa (chanusa(AT)math.washington.edu), Mar 24 2004

Keywords

• easy
• nonn
• changed

Examples

			a(3) = (3^5+(-1)^5)/2 - 2^5 - 5*(2^4-1) + 4^2 = 30.
		

References

• James Propp, Enumeration of matchings: problems and progress, pp. 255-291 in L. J. Billera et al., eds, New Perspectives in Algebraic Combinatorics, Cambridge, 1999 (see Problem 13).

Links

• James Propp, Publications and Preprints
• James Propp, Enumeration of matchings: problems and progress, in L. J. Billera et al. (eds.), New Perspectives in Algebraic Combinatorics
• Index entries for linear recurrences with constant coefficients, signature (9,-30,42,-9,-39,40,-12).

Crossrefs

Cf. A046717, A092437-A092443.

Programs

Mathematica

Table[(3^(n+2)+(-1)^(n+2))/2-2^(n+2)-(n+2)(2^(n+1)-1)+(n+1)^2,{n,0,30}] (* or *) LinearRecurrence[{9,-30,42,-9,-39,40,-12},{0,0,6,30,140,560,2058},30] (* Harvey P. Dale, Nov 27 2011 *)

Formula

a(n) = (3^(n+2)+(-1)^(n+2))/2-2^(n+2)-(n+2)*(2^(n+1)-1)+(n+1)^2.

a(n) = A092437(n, n+2), for n >= 2.

a(n) = A046717(n+2)-2^(n+2)-(n+2)*(2^(n+1)-1)+(n+1)^2.

a(n) = 9*a(n-1)-30*a(n-2)+42*a(n-3)-9*a(n-4)-39*a(n-5)+40*a(n-6)-12*a(n-7). - Harvey P. Dale, Nov 27 2011

G.f.: 2*x^2*(6*x^4-26*x^3+25*x^2-12*x+3)/((x-1)^3*(x+1)*(2*x-1)^2*(3*x-1)). - Colin Barker, Nov 22 2012

E.g.f.: exp(x)*(4*x + x^2 - 4*(2 + x)*cosh(x) - 4*(2 + x)*sinh(x) + 2*(2*cosh(x) + sinh(x))^2). - Stefano Spezia, Sep 01 2025