A254125 The number of tilings of a 4 X n rectangle using integer length rectangles with at least one side of length 1, i.e., tiles are 1 X 1, 1 X 2, ..., 1 X n, 2 X 1, 3 X 1, 4 X 1.

1, 8, 124, 2408, 50128, 1064576, 22734496, 486248000, 10404289216, 222647030144, 4764694602112, 101966374503680, 2182126445631232, 46698521255409152, 999370260391863808, 21386993399983588352, 457691719382960757760, 9794818132582234683392

Offset: 0

Author

Steve Butler, Jan 25 2015

Keywords

• nonn
• easy

Comments

Let G_n be the graph with vertices {(a,b) : 1<=a<=7, 1<=b<=2n-1, a+b odd} and edges between (a,b) and (c,d) if and only if |a-b|=|c-d|=1. Then a(n) is the number of independent sets in G_n.

Links

• Colin Barker, Table of n, a(n) for n = 0..750
• 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

Cf. A052961, A254124, A254126, A254127.

Column k=4 of A254414.

Programs

PARI

Vec((1-22*x+86*x^2-92*x^3+16*x^4)/(1-30*x+202*x^2-396*x^3 +248*x^4-32*x^5) + O(x^30)) \\ Michel Marcus, Jan 26 2015

Formula

G.f.: (1 - 22x + 86x^2 - 92x^3 + 16x^4)/(1 - 30x + 202x^2 - 396x^3 + 248x^4 - 32x^5).

a(n) = 30*a(n-1) - 202*a(n-2) + 396*a(n-3) - 248*a(n-4) + 32*a(n-5) for n>4. - Colin Barker, Jun 07 2020