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A052978
Expansion of (1-2*x)/(1-4*x-2*x^2+4*x^3).
2
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
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Independent Edge Set
Eric Weisstein's World of Mathematics, Matching
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: A374298 A268329 A223095 * A351511 A151023 A344501
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
STATUS
approved