A254151 Number of independent sets in the generalized Aztec diamond E(L_7,L_{2n-1}).

1, 16, 314, 6556, 139344, 2976416, 63663808, 1362242592, 29151501760, 623849225024, 13350628082560, 285709494797952, 6114316283697408, 130849237522680064, 2800235203724240384, 59926350645878761984, 1282452098548524184576, 27445078313878468469760

Offset: 0

Author

Steve Butler, Jan 26 2015

Keywords

• nonn

Comments

E(L_7,L_{2n-1}) is the graph with vertices {(a,b) : 1<=a<=7, 1<=b<=2n-1, a+b even} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Independent Vertex Set
• Z. Zhang, Merrifield-Simmons index of generalized Aztec diamond and related graphs, MATCH Commun. Math. Comput. Chem. 56 (2006) 625-636.
• Index entries for linear recurrences with constant coefficients, signature (30,-202,396,-248,32).

Crossrefs

Row n=4 of A331406.

Cf. A254124, A254125, A254126, A254150, A254152.

Programs

Mathematica

LinearRecurrence[{30,-202,396,-248,32},{1,16,314,6556,139344},20] (* Harvey P. Dale, May 31 2024 *)

PARI

Vec((1 - 14*x + 36*x^2 - 28*x^3 + 4*x^4)/(1 - 30*x + 202*x^2 - 396*x^3 + 248*x^4 - 32*x^5) + O(x^20)) \\ Andrew Howroyd, Jan 16 2020

Formula

Empirical g.f.: -(4*x^4-28*x^3+36*x^2-14*x+1) / (32*x^5-248*x^4+396*x^3-202*x^2+30*x-1). - Colin Barker, Jan 26 2015

The above g.f. is correct. See A331406 for bounds on the order of the recurrence. - Andrew Howroyd, Jan 16 2020

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 15 2020