A364741 Number of edge covers in the n-double cone graph.

0, 8, 160, 2009, 25872, 328208, 4165357, 52837520, 670238112, 8501756249, 107841947320, 1367938389320, 17351831692125, 220102059219128, 2791919445762040, 35414544563765129, 449221401563485632, 5698220042111151488, 72279974941308391117, 916846794068851162400, 11629888423130623254672

Offset: 0

Author

Eric W. Weisstein, Aug 05 2023

Keywords

• nonn
• easy

Comments

The n-double cone graph is defined for n >= 3. The sequence has been extended to a(0)-a(2) using the formula/recurrence. - Andrew Howroyd, Aug 08 2023

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..200
• Eric Weisstein's World of Mathematics, Double Cone Graph
• Eric Weisstein's World of Mathematics, Edge Cover
• Index entries for linear recurrences with constant coefficients, signature (13,2,-75,-17,38,-8).

Crossrefs

Cf. A000032, A206776, A297047.

Programs

Mathematica

LinearRecurrence[{13, 2, -75, -17, 38, -8}, {0, 8, 160, 2009, 25872, 328208}, 25] (* Paolo Xausa, Nov 18 2023 *)
CoefficientList[Series[-x (-8 - 56 x + 87 x^2 - 35 x^3 + 10 x^4)/((1 + x) (1 + 2 x) (1 - 3 x + x^2) (1 - 13 x + 4 x^2)), {x, 0, 20}], x] (* Eric W. Weisstein, May 25 2024 *)
Table[2 (-1)^n (2^n - 1) - LucasL[2 n] + ((13 - 3 Sqrt[17])^n + (13 + 3 Sqrt[17])^n)/2^n, {n, 20}] // Expand (* Eric W. Weisstein, May 25 2024 *)

PARI

concat(0, Vec((8 + 56*x - 87*x^2 + 35*x^3 - 10*x^4)/((1 + x)*(1 + 2*x)*(1 - 3*x + x^2)*(1 - 13*x + 4*x^2)) + O(x^20))) \\ Andrew Howroyd, Aug 08 2023

Formula

From Andrew Howroyd, Aug 08 2023: (Start)

a(n) = A206776(n)^2 - A000032(n)^2.

a(n) = 13*a(n-1) + 2*a(n-2) - 75*a(n-3) - 17*a(n-4) + 38*a(n-5) - 8*a(n-6) for n >= 6.

G.f.: x*(8 + 56*x - 87*x^2 + 35*x^3 - 10*x^4)/((1 + x)*(1 + 2*x)*(1 - 3*x + x^2)*(1 - 13*x + 4*x^2)). (End)

Extensions

a(0)-a(2) and terms a(8) and beyond from Andrew Howroyd, Aug 08 2023