A364741 -id:A364741 - cp's OEIS frontend

A372976 Number of minimum edge covers in the n-double cone graph.

0, 3, 20, 27, 64, 75, 132, 147, 224, 243, 340, 363, 480, 507, 644, 675, 832, 867, 1044, 1083, 1280, 1323, 1540, 1587, 1824, 1875, 2132, 2187, 2464, 2523, 2820, 2883, 3200, 3267, 3604, 3675, 4032, 4107, 4484, 4563, 4960, 5043, 5460, 5547, 5984, 6075, 6532, 6627, 7104

Offset: 0

Eric W. Weisstein, May 26 2024

Eric Weisstein's World of Mathematics, Double Cone Graph.
Eric Weisstein's World of Mathematics, Minimum Edge Cover.
Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Cf. A364741.

a[n_] := If[Mod[n, 2] == 0, 3*n^2 + 4*n, 3*n^2]; Table[a[n], {n, 3, 44}] (* Detlef Meya, Jun 20 2024 *)
Table[n (2 + 2 (-1)^n + 3 n), {n, 0, 20}] (* Eric W. Weisstein, Dec 09 2024 *)
LinearRecurrence[{1, 2, -2, -1, 1}, {3, 20, 27, 64, 75}, {0, 20}] (* Eric W. Weisstein, Dec 09 2024 *)
CoefficientList[Series[(x (-3 - 17 x - x^2 - 3 x^3))/((-1 + x)^3 (1 + x)^2), {x, 0, 20}], x] (* Eric W. Weisstein, Dec 09 2024 *)

a(n) = 3*n^2 + 4*n if (n mod 2 = 0), otherwise 3*n^2. - Detlef Meya, Jun 20 2024
From Eric W. Weisstein, Dec 09 2024: (Start)
G.f.: x*(-3-17*x-x^2-3*x^3)/((-1+x)^3*(1+x)^2).
a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5). (End)

a(10) and beyond from Detlef Meya, Jun 20 2024

Offset changed to 0 and a(0)-a(2) added using the formula/recurrence by Eric W. Weisstein, Dec 09 2024

A379182 Number of minimal edge covers in the n-double cone graph.

Original entry on oeis.org

0, 1, 21, 58, 149, 566, 1676, 5482, 18021, 59665, 199700, 670517, 2259384, 7624878, 25759564, 87078065, 294452965, 995889190, 3368616437, 11395096538, 38547768152, 130403228310, 441145535869, 1492374662977, 5048648849760, 17079422831941, 57779211419220, 195465558240778

Offset: 0

Eric W. Weisstein, Dec 17 2024

The sequence has been extended to n=0 using the recurrence. - Andrew Howroyd, Dec 18 2024

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, Minimal Edge Cover.
Index entries for linear recurrences with constant coefficients, signature (4,0,-2,-21,8,8,34,-12,20,-28,6,-24,20,-13,12,-4,2,-1).

Cf. A364741, A372976.

seq(n)={my(g1 = 1/(1-x -x^2 - x^3) + O(x*x^n), g2 = 1/(1-x^2-x^3) + O(x*x^n), h1 = g1 + x^2*g1 + 2*x^3*g1, h2 = g2 + x^2*g2 + 2*x^3*g2); Vec(serconvol(h1,h1) - serconvol(h2,h2) + 2*serconvol(h2, x*deriv(2*x^2*g2 + x^3*g2)), -n-1)} \\ Andrew Howroyd, Dec 18 2024

G.f.: x*(1 + 17*x - 26*x^2 - 81*x^3 + 33*x^4 - 39*x^5 + 118*x^6 - 312*x^7 + 461*x^8 - 260*x^9 + 183*x^10 - 211*x^11 + 33*x^12 + 10*x^13 + 3*x^14 - 2*x^16)/((1 + x + x^2 - x^3)*(1 + x - x^3)^2*(1 - 2*x + x^2 - x^3)^2*(1 - 3*x - x^2 - x^3)). - Andrew Howroyd, Dec 18 2024

a(0)-a(2) prepended and a(8) onwards from Andrew Howroyd, Dec 18 2024

cp's OEIS Frontend

A372976 Number of minimum edge covers in the n-double cone graph.

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