A367967 a(n) = 3/4*(3^(n + 1) - 2*n - 4*n^2 - 3).

0, 0, 3, 27, 126, 462, 1521, 4761, 14556, 44028, 132543, 398199, 1195290, 3586698, 10761069, 32284341, 96854328, 290564472, 871695099, 2615087187, 7845263670, 23535793350, 70607382633, 211822150737, 635466455316, 1906399369332, 5719198111671, 17157594338991

Offset: 0

Author

Eric W. Weisstein, Dec 06 2023

Keywords

• nonn
• easy

Comments

Number of 5-cycles in the n-Dorogovtsev-Goltsev-Mendes graph (using the convention that DGM(0) = P_2).

Links

• Eric Weisstein's World of Mathematics, Dorogovtsev-Goltsev-Mendes Graph.
• Eric Weisstein's World of Mathematics, Graph Cycle.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,10,-3).

Crossrefs

Cf. A003462(n) (3-cycles), A290764(n-1) (4-cycles), A367968(n) (6-cycles).

Programs

Mathematica

Table[3/4 (3^(n + 1) - 2 n - 4 n^2 - 3), {n, 0, 20}]
LinearRecurrence[{6, -12, 10, -3}, {0, 0, 3, 27}, 20]
CoefficientList[Series[3 x^2 (1 + 3 x)/((-1 + x)^3 (-1 + 3 x)), {x, 0, 20}], x]

Formula

a(n) = 3/4*(3^(n + 1) - 2*n - 4*n^2 - 3).

a(n) = 6*a(n-1) - 12*a(n-2) + 10*a(n-3) - 3*a(n-4).

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