A370846 a(n) = 5 + 4^n + 3*((-i*sqrt(7) - 3)^n + (i*sqrt(7) - 3)^n)/2^n.

0, 24, 96, 168, 1200, 3960, 16128, 66888, 259152, 1052184, 4195488, 16759272, 67158000, 268359864, 1073772096, 4295178888, 17179113360, 68720897880, 274876666848, 1099509663528, 4398057364272, 17592161341944, 70368774872448, 281474983436232, 1125899763886800

Offset: 1

Author

Eric W. Weisstein, Mar 03 2024

Keywords

• nonn
• easy

Comments

For n > 2 and n != 0 (mod 3), also the number of minimum vertex colorings in the n-antiprism graph.

Links

• Eric Weisstein's World of Mathematics, Antiprism Graph.
• Eric Weisstein's World of Mathematics, Minimum Vertex Coloring.
• Index entries for linear recurrences with constant coefficients, signature (2,7,8,-16).

Programs

Mathematica

Table[5 + 4^n + 3 ((-I Sqrt[7] - 3)^n + (I Sqrt[7] - 3)^n)/2^n, {n, 25}]
LinearRecurrence[{2, 7, 8, -16}, {0, 24, 96, 168}, 20]
CoefficientList[Series[-(24 x (-1 + 2 x) (1 + 4 x)/((-1 + x) (-1 + 4 x) (1 + 3 x + 4 x^2))), {x, 0, 20}], x]

Formula

a(n) = 5 + A000302(n) + 3*A128415(n).

a(n) = 2*a(n-1) + 7*a(n-2) + 8*a(n-3) - 16*a(n-4).

G.f.: -24*x^2*(-1+2*x)*(1+4*x)/((-1+x)*(-1+4*x)*(1+3*x+4*x^2)).