A294152 Chromatic invariant of the n-antiprism graph.

0, 2, 11, 38, 112, 309, 828, 2190, 5759, 15106, 39580, 103657, 271416, 710618, 1860467, 4870814, 12752008, 33385245, 87403764, 228826086, 599074535, 1568397562, 4106118196, 10749957073, 28143753072, 73681302194, 192900153563, 505019158550, 1322157322144

Offset: 1

Author

Eric W. Weisstein, Nov 16 2017

Keywords

• nonn
• easy

Comments

Extended to a(1)-a(2) using the formula/recurrence.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Antiprism Graph
• Eric Weisstein's World of Mathematics, Chromatic Invariant
• Index entries for linear recurrences with constant coefficients, signature (5, -8, 5, -1).

Crossrefs

Cf. A005248 (lucasl(2*n)).

Programs

Mathematica

Table[LucasL[2 n] - 2 n - 1, {n, 3, 20}]
LinearRecurrence[{5, -8, 5, -1}, {0, 2, 11, 38}, 20]
CoefficientList[Series[(x (2 + x - x^2))/((-1 + x)^2 (1 - 3 x + x^2)), {x, 0, 20}], x]

PARI

concat(0, Vec(x^2*(2 + x - x^2)/((-1 + x)^2*(1 - 3*x + x^2)) + O(x^40))) \\ Colin Barker, Nov 16 2017

Formula

a(n) = A005248(n) - 2*n - 1.

a(n) = phi^(2*n) + phi^(-2*n) - 2*n - 1.

a(n) = 5*a(n-1) - 8*a(n-2) + 5*a(n-3) - a(n-4).

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

a(n) = -1 + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n - 2*n. - Colin Barker, Nov 16 2017