A121124 Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).

1, 4, 21, 138, 864, 5526, 34992, 221724, 1399680, 8818632, 55427328, 347684400, 2176782336, 13604912928, 84894511104, 528958247616, 3291294892032, 20453047668864, 126949945835520, 787089669219072, 4874877920083968, 30163307160752640, 186464080443211776, 1151689908801235968

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=1.
• Index entries for linear recurrences with constant coefficients, signature (12,-30,-72,216).

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
A121124 := proc(r::integer)
    q := 9 ;
    a := 1 ;
    Jra(r,a,q)+binomial(2,r-a)+( 1 +(-1)^(r+a) +(1+(-1)^a)*(1-(-1)^r)*floor((q-3)/2)/2)*Hra(floor(r/2),floor(a/2),q) ;
    %/4 ;
end proc:
seq(A121124(n),n=2..30)  # R. J. Mathar, Aug 01 2019

Mathematica

Join[{1, 4}, LinearRecurrence[{12, -30, -72, 216}, {21, 138, 864, 5526}, 22]] (* Jean-François Alcover, Apr 04 2020 *)

Formula

From R. J. Mathar, Aug 01 2019: (Start)

G.f.: x^2 +4*x^3 -3*x^4*(7-38*x-54*x^2+270*x^3) / ( (6*x^2-1)*(-1+6*x)^2 ).

a(n) = A000400((n-1)/2)/12 +6^(n-1)/16 +A053469(n+1)/864, where Axxxxx(.) is zero for fractional indices, n>3. (End)