A297047 -id:A297047 - cp's OEIS frontend

A356213 Number of edge covers in the n-trapezohedral graph.

4, 104, 1699, 23904, 317044, 4101107, 52473796, 668177568, 8490113467, 107776172264, 1367566963756, 17349734444643, 220090218116188, 2791852592070632, 35414167120396459, 449219270600324928, 5698208011194600148, 72279907017666274643, 916846410588661477204

Offset: 1

Eric W. Weisstein, Jul 29 2022

nonn

Sequence extended to n = 1 using the formula/recurrence.

Christian Sievers, Table of n, a(n) for n = 1..200
Eric Weisstein's World of Mathematics, Edge Cover
Eric Weisstein's World of Mathematics, Trapezohedral Graph
Index entries for linear recurrences with constant coefficients, signature (22,-142,321,-242,74,-8).

Cf. A297047.

Table[LucasL[2 n] - 2 ((3 - Sqrt[7])^n + (3 + Sqrt[7])^n) + ((13 - 3 Sqrt[17])^n + (13 + 3 Sqrt[17])^n)/2^n, {n, 20}] // Expand (* Eric W. Weisstein, May 26 2024 *)
LinearRecurrence[{22, -142, 321, -242, 74, -8}, {4, 104, 1699, 23904, 317044, 4101107}, 20] (* Eric W. Weisstein, May 26 2024 *)
CoefficientList[Series[-(((-2 + x) (2 + 9 x - 6 x^2 + 2 x^3))/((1 - 3 x + x^2) (1 - 6 x + 2 x^2) (1 - 13 x + 4 x^2))), {x, 0, 20}], x] (* Eric W. Weisstein, May 26 2024 *)

a(n) = 22*a(n-1) - 142*a(n-2) + 321*a(n-3) - 242*a(n-4) + 74*a(n-5) - 8*a(n-6) for n > 6.

G.f.: -(-2+x)*x*(2+9*x-6*x^2+2*x^3)/((1-3*x+x^2)*(1-6*x+2*x^2)*(1-13*x+4*x^2)).

a(9)-a(12) from Andrew Howroyd, Jan 27 2023
More terms from Christian Sievers, Nov 20 2023
a(1)-a(2) prepended by Eric W. Weisstein, May 26 2024
Offset updated for a(1)-a(2) by Sean A. Irvine, Aug 11 2024

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

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Aug 05 2023

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

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).

Cf. A000032, A206776, A297047.

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 *)

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

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)

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

cp's OEIS Frontend

A356213 Number of edge covers in the n-trapezohedral graph.

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