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

%I A121123 #21 Mar 31 2020 10:53:47
%S A121123 1,3,12,63,342,1998,11772,70308,420552,2521368,15120432,90710928,
%T A121123 544218912,3265243488,19591180992,117546666048,705278316672,
%U A121123 4231667380608,25389994205952,152339950119168,914039640248832,5484237750793728,32905426141965312,197432556307596288
%N A121123 Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).
%H A121123 J. Brunvoll, S. J. Cyvin and B. N. Cyvin, <a href="https://doi.org/10.1016/0166-1280(95)04463-9">Isomer enumeration of polygonal systems...</a>, J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 12, q=9, alpha=0.
%H A121123 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,6,-36).
%F A121123 From _Colin Barker_, Aug 30 2013: (Start)
%F A121123 a(n) = 6*a(n-1)+6*a(n-2)-36*a(n-3) for n>5.
%F A121123 G.f.: x^2 -3*x^3*(-1+2*x+9*x^2) / ( (6*x-1)*(6*x^2-1) ). (End)
%F A121123 a(n) = A026532(n+1)/12 +6^(n-2)/4, n>2. - _R. J. Mathar_, Aug 01 2019
%p A121123 # Exhibit 1
%p A121123 Hra := proc(r::integer,a::integer,q::integer)
%p A121123     binomial(r-1,a-1)*(q-3)+binomial(r-1,a) ;
%p A121123     %*(q-3)^(r-a-1) ;
%p A121123 end proc:
%p A121123 Jra := proc(r::integer,a::integer,q::integer)
%p A121123     binomial(r-2,a-2)*(q-3)^2 +2*binomial(r-2,a-1)*(q-3) +binomial(r-2,a) ;
%p A121123     %*(q-3)^(r-a-2) ;
%p A121123 end proc:
%p A121123 # Exhibit 2, I_m
%p A121123 A121123 := proc(r::integer)
%p A121123     local q,a,f ;
%p A121123     q := 9 ;
%p A121123     a := 0 ;
%p A121123     f := 1 +(-1)^(r+a) +(1+(-1)^a) *(1-(-1)^r) *floor((q-3)/2) /2 ;
%p A121123     Jra(r,a,q)+binomial(2,r-a)+f*Hra(floor(r/2),floor(a/2),q) ;
%p A121123     %/4 ;
%p A121123 end proc:
%p A121123 seq(A121123(n),n=2..30) ; # _R. J. Mathar_, Aug 01 2019
%t A121123 Join[{1}, LinearRecurrence[{6, 6, -36}, {3, 12, 63}, 23]] (* _Jean-François Alcover_, Mar 31 2020 *)
%K A121123 nonn,easy
%O A121123 2,2
%A A121123 _N. J. A. Sloane_, Aug 13 2006