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

Author

Eric W. Weisstein, Jun 03 2017

Keywords

• nonn

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).

Crossrefs

Cf. A077263, A124352, A124353, A287992.

Programs

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]

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

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)

Extensions

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