A297384 Number of Eulerian cycles in the n-antiprism graph.

4, 44, 372, 2932, 22484, 170196, 1279828, 9590612, 71736660, 536055124, 4003591508, 29892900180, 223162389844, 1665861735764, 12434781197652, 92816950121812, 692805066118484, 5171207088198996, 38598573880071508, 288104312443589972, 2150442403051689300

Offset: 1

Author

Eric W. Weisstein, Dec 29 2017

Keywords

• nonn
• easy

Comments

Sequence extrapolated to n=1 and n=2 using the recurrence. - Andrew Howroyd, Jan 11 2018

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, Eulerian Cycle
• Index entries for linear recurrences with constant coefficients, signature (13, -48, 52, -16).

Programs

Mathematica

Table[2^n ((2 - Sqrt[3])^n + (2 + Sqrt[3])^n) - 2/3 (2 + 4^n), {n, 20}] // Expand
Table[2^n LucasL[2 n, Sqrt[2]] - 2/3 (2 + 4^n), {n, 20}] // Round
LinearRecurrence[{13, -48, 52, -16}, {4, 44, 372, 2932}, 20]
CoefficientList[Series[-4 (-1 + 2 x + 2 x^2)/(1 - 13 x + 48 x^2 - 52 x^3 + 16 x^4), {x, 0, 20}], x]

PARI

Vec(4*(1 - 2*x - 2*x^2)/((1 - 8*x + 4*x^2)*(1 - 4*x)*(1 - x)) + O(x^30)) \\ Andrew Howroyd, Jan 11 2018

Formula

From Andrew Howroyd, Jan 11 2018: (Start)

a(n) = 13*a(n-1) - 48*a(n-2) + 52*a(n-3) - 16*a(n-4).

G.f.: 4*x*(1 - 2*x - 2*x^2)/((1 - 8*x + 4*x^2)*(1 - 4*x)*(1 - x)).

(End)

From Eric W. Weisstein, Jan 12 2018: (Start)

a(n) = 2^n*((2 - sqrt(3))^n + (2 + sqrt(3))^n) - 2/3*(2 + 4^n).

a(n) = 2^n*A003500(n) - 2/3*(2 + 4^n).

a(n) = A003500(n) - A039301(n-1).

(End)