A030517 Number of walks of length n between two vertices on an icosahedron at distance 1.

1, 2, 13, 52, 273, 1302, 6573, 32552, 163073, 813802, 4070573, 20345052, 101733073, 508626302, 2543170573, 12715657552, 63578483073, 317891438802, 1589458170573, 7947285970052, 39736434733073, 198682149251302, 993410770670573, 4967053731282552

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• walk
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,10,-20,-25).

Crossrefs

Cf. A030518.

Programs

Mathematica

LinearRecurrence[{4, 10, -20, -25}, {1, 2, 13, 52}, 24] (* Jean-François Alcover, Jul 12 2021 *)

PARI

Vec(x*(1-2*x-5*x^2)/((1+x)*(1-5*x)*(1-5*x^2)) + O(x^30)) \\ Colin Barker, Oct 17 2016

Formula

a(n) = 2*a(n-1) + 2*A030518(n-1) + 5*a(n-2).

A030518(n) = 2*a(n-1) + 2*A030518(n-1) + 5*A030518(n-2).

From Emeric Deutsch, Apr 03 2004: (Start)

a(n) = 5^n/12 - (-1)^n/12 + (sqrt(5))^(n+1)/20 + (-sqrt(5))^(n+1)/20.

a(n) = 4*a(n-1) + 10*a(n-2) - 20*a(n-3) - 25*a(n-4) for n>=5. (End)

From Colin Barker, Oct 17 2016: (Start)

G.f.: x*(1 - 2*x - 5*x^2)/((1 + x)*(1 - 5*x)*(1 - 5*x^2)).

a(n) = (5^n - 1)/12 for n even.

a(n) = (6*5^((n-1)/2) + 5^n + 1)/12 for n odd. (End)