A121123 Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).

1, 3, 12, 63, 342, 1998, 11772, 70308, 420552, 2521368, 15120432, 90710928, 544218912, 3265243488, 19591180992, 117546666048, 705278316672, 4231667380608, 25389994205952, 152339950119168, 914039640248832, 5484237750793728, 32905426141965312, 197432556307596288

Offset: 2

Author

N. J. A. Sloane, Aug 13 2006

Keywords

• nonn
• easy

Links

• J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 12, q=9, alpha=0.
• Index entries for linear recurrences with constant coefficients, signature (6,6,-36).

Programs

Maple

# Exhibit 1
Hra := proc(r::integer,a::integer,q::integer)
    binomial(r-1,a-1)*(q-3)+binomial(r-1,a) ;
    %*(q-3)^(r-a-1) ;
end proc:
Jra := proc(r::integer,a::integer,q::integer)
    binomial(r-2,a-2)*(q-3)^2 +2*binomial(r-2,a-1)*(q-3) +binomial(r-2,a) ;
    %*(q-3)^(r-a-2) ;
end proc:
# Exhibit 2, I_m
A121123 := proc(r::integer)
    local q,a,f ;
    q := 9 ;
    a := 0 ;
    f := 1 +(-1)^(r+a) +(1+(-1)^a) *(1-(-1)^r) *floor((q-3)/2) /2 ;
    Jra(r,a,q)+binomial(2,r-a)+f*Hra(floor(r/2),floor(a/2),q) ;
    %/4 ;
end proc:
seq(A121123(n),n=2..30) ; # R. J. Mathar, Aug 01 2019

Mathematica

Join[{1}, LinearRecurrence[{6, 6, -36}, {3, 12, 63}, 23]] (* Jean-François Alcover, Mar 31 2020 *)

Formula

From Colin Barker, Aug 30 2013: (Start)

a(n) = 6*a(n-1)+6*a(n-2)-36*a(n-3) for n>5.

G.f.: x^2 -3*x^3*(-1+2*x+9*x^2) / ( (6*x-1)*(6*x^2-1) ). (End)

a(n) = A026532(n+1)/12 +6^(n-2)/4, n>2. - R. J. Mathar, Aug 01 2019