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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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified November 30 12:40 EST 2022. Contains 358441 sequences. (Running on oeis4.)