OFFSET
1,3
COMMENTS
Numbers of polypentagons with two connected internal vertices.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 10.
S. J. Cyvin et al., Theory of polypentagons, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
Index entries for linear recurrences with constant coefficients, signature (4,-2,-8,8).
FORMULA
G.f.: x*(2*x^6+2*x^5-5*x^3+3*x-1)/((2*x-1)^2*(2*x^2-1)). - Colin Barker, Nov 30 2012
From Colin Barker, Oct 28 2016: (Start)
a(n) = 2^(n-5)*(n+2) for n>3 and even.
a(n) = 2^(n-5)*(n+2)+2^((n-5)/2) for n>3 and odd.
a(n) = 4*a(n-1)-2*a(n-2)-8*a(n-3)+8*a(n-4) for n>7.
(End)
MAPLE
H := proc(r, alpha, q) local rhalf, alphahalf ; rhalf := floor(r/2) ; alphahalf := floor(alpha/2) ; (binomial(rhalf-1, alphahalf-1)*(q-3)+binomial(rhalf-1, alphahalf))*(q-3)^(rhalf-alphahalf-1) ; end:
J := proc(r, alpha, q) (binomial(r-2, alpha-2)*(q-3)^2+2*binomial(r-2, alpha-1)*(q-3)+binomial(r-2, alpha))*(q-3)^(r-alpha-2) ; end: Ifunc := proc(r, alpha, q) J(r, alpha, q)/4+binomial(2, r-alpha)/4+ (1+(-1)^(r+alpha)+(1+(-1)^alpha)*(1-(-1)^r)/2)*H(r, alpha, q)/4 ; end:
A121133 := proc(n) if n = 1 then 1; else Ifunc(n, 1, 5) ; fi ; end: for n from 1 to 80 do printf("%d, ", A121133(n)) ; od: # R. J. Mathar, Aug 07 2008
MATHEMATICA
Rest@ CoefficientList[Series[x (2 x^6 + 2 x^5 - 5 x^3 + 3 x - 1)/((2 x - 1)^2*(2 x^2 - 1)), {x, 0, 34}], x] (* Michael De Vlieger, Oct 28 2016 *)
LinearRecurrence[{4, -2, -8, 8}, {1, 1, 2, 3, 8, 16, 38}, 40] (* Harvey P. Dale, Feb 10 2019 *)
PROG
(PARI) Vec(x*(2*x^6+2*x^5-5*x^3+3*x-1)/((2*x-1)^2*(2*x^2-1)) + O(x^50)) \\ Colin Barker, Oct 28 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Aug 13 2006, Sep 24 2006
EXTENSIONS
Edited and extended by R. J. Mathar, Aug 07 2008
STATUS
approved