A287992 Number of (undirected) paths in the prism graph Y_n.

1, 26, 129, 444, 1285, 3366, 8281, 19544, 44829, 100770, 223201, 488916, 1061749, 2289854, 4910505, 10480176, 22275661, 47178234, 99605809, 209704940, 440390181, 922733526, 1929364729, 4026514824, 8388588925, 17448283346, 36238762881, 75161901444, 155692535509, 322122515310

Offset: 1

Author

Eric W. Weisstein, Jun 04 2017

Keywords

• nonn
• easy

Comments

Extended to a(1)-a(2) using the formula.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Eric Weisstein's World of Mathematics, Graph Path.
• Eric Weisstein's World of Mathematics, Prism Graph.
• Index entries for linear recurrences with constant coefficients, signature (8,-26,44,-41,20,-4).

Crossrefs

Cf. A077265, A124350, A124349, A287988, A020874, A288516.

Programs

Mathematica

Table[(5 2^(n + 1) - 5 n - n^2 - 13) n, {n, 20}]
LinearRecurrence[{8, -26, 44, -41, 20, -4}, {1, 26, 129, 444, 1285, 3366}, 20]
CoefficientList[Series[(1 + 18 x - 53 x^2 + 44 x^3 - 16 x^4)/((1 - x)^4 (1 - 2 x)^2), {x, 0, 20}], x]

PARI

Vec(x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2) + O(x^30)) \\ Colin Barker, Jun 04 2017

Formula

a(n) = (5*2^(n + 1) - 5*n - n^2 - 13)*n.

From Colin Barker, Jun 04 2017: (Start)

G.f.: x*(1 + 18*x - 53*x^2 + 44*x^3 - 16*x^4) / ((1 - x)^4*(1 - 2*x)^2).

a(n) = 8*a(n-1) - 26*a(n-2) + 44*a(n-3) - 41*a(n-4) + 20*a(n-5) - 4*a(n-6) for n>6. (End)