A122679 Invariant number of internal vertices of n-circum-C_5 H_5 systems.

0, 5, 25, 80, 225, 605, 1600, 4205, 11025, 28880, 75625, 198005, 518400, 1357205, 3553225, 9302480, 24354225, 63760205, 166926400, 437019005, 1144130625, 2995372880, 7841988025, 20530591205, 53749785600, 140718765605, 368406511225, 964500768080

Offset: 1

Author

N. J. A. Sloane, Sep 23 2006

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Azulenoids, MATCH, No. 34, 1996, 91-108.
• Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Programs

Mathematica

LinearRecurrence[{4,-4,1},{0,5,25},40] (* Harvey P. Dale, Apr 21 2015 *)

PARI

concat(0, Vec(-5*x^2*(1+x)/((x-1)*(x^2-3*x+1)) + O(x^40))) \\ Colin Barker, Nov 03 2016

Formula

a(n) = 15*Fibonacci(2*k-1)-5*Fibonacci(2*k)-10 = 5*A004146(n-1).

G.f.: -5*x^2*(1+x) / ( (x-1)*(x^2-3*x+1) ). - R. J. Mathar, Nov 23 2014

a(1)=0, a(2)=5, a(3)=25, a(n) = 4*a(n-1)-4*a(n-2)+a(n-3). - Harvey P. Dale, Apr 21 2015

a(n) = -5*2^(-1-n)*(2^(2+n)-(3-sqrt(5))^n*(3+sqrt(5))+(-3+sqrt(5))*(3+sqrt(5))^n). - Colin Barker, Nov 03 2016

Extensions

More terms from Harvey P. Dale, Apr 21 2015