A122678 Invariant number of polygons of n-circum-C_5 H_5 systems.

1, 6, 21, 61, 166, 441, 1161, 3046, 7981, 20901, 54726, 143281, 375121, 982086, 2571141, 6731341, 17622886, 46137321, 120789081, 316229926, 827900701, 2167472181, 5674515846, 14856075361, 38893710241, 101825055366, 266581455861, 697919312221, 1827176480806

Offset: 1

Author

N. J. A. Sloane, Sep 23 2006

Keywords

• nonn
• easy

Comments

Note that the index in Eq. (5) of the paper must be 2k-1, not 2k as published, to reproduce the numbers. - R. J. Mathar, Nov 23 2014

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

PARI

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

Formula

a(n) = 5*Fibonacci(2*k-1)-4 = A106729(k-1)-4.

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

From Colin Barker, Nov 03 2016: (Start)

a(n) = 2^(-1-n)*(-2^(3+n)-(-5+sqrt(5))*(3+sqrt(5))^n+(3-sqrt(5))^n*(5+sqrt(5))).

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