A287988 Number of (undirected) paths in the n-antiprism graph.

2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640, 31292844, 115494312, 423283562, 1542120664, 5589611460, 20170172896, 72499928322, 259692909048, 927342338956, 3302291258200, 11730149911914, 41572470711288, 147031327493572, 519029653663056

Offset

1,1

Comments

Sequence extrapolated to n=1 using recurrence. - Andrew Howroyd, Jun 05 2017

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Antiprism Graph

Eric Weisstein's World of Mathematics, Graph Path

Index entries for linear recurrences with constant coefficients, signature (10, -37, 64, -58, 36, -26, 16, -5, 2, -1).

Formula

From Andrew Howroyd, Jun 05 2017 (Start)

a(n) = 10*a(n-1)-37*a(n-2)+64*a(n-3) -58*a(n-4)+36*a(n-5)-26*a(n-6) +16*a(n-7)-5*a(n-8) +2*a(n-9)-a(n-10) for n>10.

G.f.: 2*x*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1) * (1-2*x-x^2) / ((1-x)^4 * (1-3*x-x^2-x^3)^2).

(End)

Mathematica

Table[n RootSum[-1 - # - 3 #^2 + #^3 &, 23 #^n + 32 #^(n + 1) + 5 #^(n + 2) &]/44 - 7 n - 3 n^2 - 2 n^3, {n, 20}]
LinearRecurrence[{10, -37, 64, -58, 36, -26, 16, -5, 2, -1}, {2, 56, 396, 2040, 9130, 37944, 151172, 586608, 2235618, 8407640}, 20]
CoefficientList[Series[(2 (2 x^6 + 4 x^5 + x^4 + 24 x^3 - 4 x^2 + 20 x + 1) (1 - 2 x - x^2))/((1 - x)^4 (1 - 3 x - x^2 - x^3)^2), {x, 0, 20}], x]

Prog

(PARI)
Vec(2*(2*x^6+4*x^5+x^4+24*x^3-4*x^2+20*x+1)*(1-2*x-x^2)/((1-x)^4*(1-3*x-x^2-x^3)^2) + O(x^20)) \\ Andrew Howroyd, Jun 05 2017

Crossrefs

Cf. A077263, A124352, A124353, A287992.

Sequence in context: A037176 A045819 A281198 * A193830 A196539 A056046

Adjacent sequences: A287985 A287986 A287987 * A287989 A287990 A287991

Keyword

nonn

Author

Eric W. Weisstein, Jun 03 2017

Extensions

a(1)-a(2) and a(14)-a(24) from Andrew Howroyd, Jun 05 2017

Status

approved