A121133 Numbers of isomers of unbranched a-4-catapolypentagons - see Brunvoll reference for precise definition.

1, 1, 2, 3, 8, 16, 38, 80, 180, 384, 840, 1792, 3856, 8192, 17440, 36864, 77888, 163840, 344192, 720896, 1507584, 3145728, 6554112, 13631488, 28312576, 58720256, 121636864, 251658240, 520097792, 1073741824, 2214600704, 4563402752, 9395257344, 19327352832

Offset: 1

Author

N. J. A. Sloane, Aug 13 2006, Sep 24 2006

Keywords

• nonn
• easy

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).

Programs

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 *)

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

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)

Extensions

Edited and extended by R. J. Mathar, Aug 07 2008