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

%I A121133 #29 Jun 20 2022 08:46:34
%S A121133 1,1,2,3,8,16,38,80,180,384,840,1792,3856,8192,17440,36864,77888,
%T A121133 163840,344192,720896,1507584,3145728,6554112,13631488,28312576,
%U A121133 58720256,121636864,251658240,520097792,1073741824,2214600704,4563402752,9395257344,19327352832
%N A121133 Numbers of isomers of unbranched a-4-catapolypentagons - see Brunvoll reference for precise definition.
%C A121133 Numbers of polypentagons with two connected internal vertices.
%H A121133 Colin Barker, <a href="/A121133/b121133.txt">Table of n, a(n) for n = 1..1000</a>
%H A121133 J. Brunvoll, S. J. Cyvin and B. N. Cyvin, <a href="http://dx.doi.org/10.1016/0166-1280(95)04463-9">Isomer enumeration of polygonal systems...</a>, J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 10.
%H A121133 S. J. Cyvin et al., <a href="http://dx.doi.org/10.1021/ci00013a027">Theory of polypentagons</a>, J. Chem. Inf. Comput. Sci., 33 (1993), 466-474.
%H A121133 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-8,8).
%F A121133 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
%F A121133 From _Colin Barker_, Oct 28 2016: (Start)
%F A121133 a(n) = 2^(n-5)*(n+2) for n>3 and even.
%F A121133 a(n) = 2^(n-5)*(n+2)+2^((n-5)/2) for n>3 and odd.
%F A121133 a(n) = 4*a(n-1)-2*a(n-2)-8*a(n-3)+8*a(n-4) for n>7.
%F A121133 (End)
%p A121133 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:
%p A121133 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:
%p A121133 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
%t A121133 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 *)
%t A121133 LinearRecurrence[{4,-2,-8,8},{1,1,2,3,8,16,38},40] (* _Harvey P. Dale_, Feb 10 2019 *)
%o A121133 (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
%K A121133 nonn,easy
%O A121133 1,3
%A A121133 _N. J. A. Sloane_, Aug 13 2006, Sep 24 2006
%E A121133 Edited and extended by _R. J. Mathar_, Aug 07 2008