A320346 a(n) is the number of perfect matchings in the graph with vertices labeled 1 to 2n with edges {i,j} for 1<=|i-j|<=4.

1, 1, 3, 12, 35, 105, 329, 1014, 3116, 9610, 29625, 91279, 281303, 866948, 2671727, 8233671, 25374513, 78198928, 240992592, 742688720, 2288811009, 7053635369, 21737825143, 66991419284, 206453506615, 636246416105, 1960778041673, 6042706771910, 18622355183932, 57390193784986, 176864543185497

Offset: 0

Author

Robert Israel, Jan 22 2019

Keywords

• nonn

Examples

			The a(3) = 12 matchings are (12)(34)(56), (12)(35)(46), (12)(36)(45), (13)(24)(56), (13)(25)(46), (13)(26)(45), (14)(23)(56), (14)(25)(36), (14)(26)(35), (15)(23)(46), (15)(24)(36), (15)(26)(34).
		

Links

• Robert Israel, Table of n, a(n) for n = 0..2044
• M. Schwartz, Efficiently computing the permanent and Hafnian of some banded Toeplitz matrices, Linear Algebra and its Applications 430 (2009), 1364-1374.
• Index entries for linear recurrences with constant coefficients, signature (2,1,6,3,2,1,-2,-1).

Crossrefs

Cf. A052967.

Programs

Maple

f:= gfun:-rectoproc({a(n) + 2*a(n + 1) - a(n + 2) - 2*a(n + 3) - 3*a(n + 4) - 6*a(n + 5)- a(n + 6) - 2*a(n + 7) + a(n + 8), a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 12, a(4) = 35, a(5) = 105, a(6) = 329, a(7) = 1014}, a(n), remember):
map(f, [$0..100]);

Mathematica

LinearRecurrence[{2, 1, 6, 3, 2, 1, -2, -1}, {1, 1, 3, 12, 35, 105, 329, 1014}, 40] (* Jean-François Alcover, Apr 30 2019 *)

Formula

G.f.: (-x^4 - x^3 - x + 1)/(1 - 2*x - x^2 - 6*x^3 - 3*x^4 - 2*x^5 - x^6 + 2*x^7 + x^8).

Extensions

a(0)=1 prepended and edited by Alois P. Heinz, Feb 28 2019