A052978 Expansion of (1-2*x)/(1-4*x-2*x^2+4*x^3).

1, 2, 10, 40, 172, 728, 3096, 13152, 55888, 237472, 1009056, 4287616, 18218688, 77413760, 328941952, 1397720576, 5939111168, 25236118016, 107231812096, 455643039744, 1936091311104, 8226724075520, 34956506765312, 148535109967872, 631146557100032

Offset

0,2

Comments

a(n) = element(1,3) in A^(n+1), where A is the 5 X 5 matrix:

[1, 1, 1, 1, 1]

[1, 1, 0, 1, 1]

[1, 0, 0, 0, 1]

[1, 1, 0, 1, 1]

[1, 1, 1, 1, 1]. - Lechoslaw Ratajczak, May 03 2017

Also the number of matchings in the 2 X n king graph for n >= 1. - Eric W. Weisstein, Oct 03 2017

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1050

Eric Weisstein's World of Mathematics, Grid Graph

Eric Weisstein's World of Mathematics, Independent Edge Set

Eric Weisstein's World of Mathematics, Matching

Index entries for linear recurrences with constant coefficients, signature (4,2,-4).

Formula

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

Recurrence: {a(0)=1, a(1)=2, a(2)=10, 4*a(n)-2*a(n+1)-4*a(n+2)+a(n+3)=0.}

a(n) = Sum(-1/158*(-11-42*r+24*r^2)*r^(-1-n) where r=RootOf(1-4*_Z-2*_Z^2+4*_Z^3))

Maple

spec := [S, {S=Sequence(Prod(Union(Sequence(Union(Z, Z)), Z), Union(Z, Z)))}, unlabeled ]: seq(combstruct[count ](spec, size=n), n=0..20);

Mathematica

LinearRecurrence[{4, 2, -4}, {1, 2, 10}, 40] (* Vincenzo Librandi, Jun 23 2012 *)
Table[RootSum[4 - 2 # - 4 #^2 + #^3 &, 30 #^n - 13 #^(n + 1) + 6 #^(n + 2) &]/158, {n, 0, 20}] (* Eric W. Weisstein, Oct 03 2017 *)
Table[RootSum[1 - 4 # - 2 #^2 + 4 #^3 &, (11 + 42 # - 24 #^2)/#^(n + 1) &]/158, {n, 0, 20}] (* Eric W. Weisstein, Oct 03 2017 *)
CoefficientList[Series[(1 - 2 x)/(1 - 4 x - 2 x^2 + 4 x^3), {x, 0, 20}], x] (* Eric W. Weisstein, Oct 03 2017 *)

Prog

(Magma) I:=[1, 2, 10]; [n le 3 select I[n] else 4*Self(n-1)+2*Self(n-2)-4*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012
(PARI) Vec((1-2*x)/(1-4*x-2*x^2+4*x^3) + O(x^30)) \\ Michel Marcus, May 06 2017

Crossrefs

Sequence in context: A193519 A268329 A223095 * A351511 A151023 A344501

Adjacent sequences: A052975 A052976 A052977 * A052979 A052980 A052981

Keyword

easy,nonn

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Status

approved