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A121138
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Numbers of isomers of unbranched a-4-catapolyheptagons - see Brunvoll reference for precise definition.
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2
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1, 1, 3, 10, 46, 192, 840, 3584, 15392, 65536, 278656, 1179648, 4981248, 20971520, 88082432, 369098752, 1543512064, 6442450944, 26843578368, 111669149696, 463856599040, 1924145348608, 7971459825664, 32985348833280, 136339443941376, 562949953421312
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OFFSET
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1,3
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REFERENCES
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J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13.
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LINKS
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FORMULA
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Empirical (for n>=4): a(n) = 2^(n-8)*((n+6)*2^n + 8 - 8*(-1)^n). - Vaclav Kotesovec, Nov 29 2012
Empirical G.f.: -3/128 + x^3/4 + x^2/2 + 49*x/64 - (296*x^3 - 140*x^2 + 6*x + 3)/(128*(2*x-1)*(2*x+1)*(4*x-1)^2). - Vaclav Kotesovec, Nov 29 2012
a(n) = 4^(n-4)*(n+6) for n>3 and even.
a(n) = 2^(n-8)*(2^n*(n+6)+16) for n>3 and odd.
a(n) = 8*a(n-1)-12*a(n-2)-32*a(n-3)+64*a(n-4) for n>7.
(End)
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MAPLE
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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: A121138 := proc(n) if n = 1 then 1 ; else Ifunc(n, 1, 7) ; fi ; end: for n from 1 to 80 do printf("%d, ", A121138(n)) ; od: # R. J. Mathar, Aug 07 2008
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MATHEMATICA
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Rest[CoefficientList[Series[-3/128+x^3/4+x^2/2+49*x/64-(296*x^3-140*x^2+6*x+3)/(128*(2*x-1)*(2*x+1)*(4*x-1)^2), {x, 0, 20}], x]] (* Vaclav Kotesovec, Nov 29 2012 *)
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PROG
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(PARI) Vec(x*(1-7*x+7*x^2+30*x^3-30*x^4-24*x^5-16*x^6)/((1-2*x)*(1+2*x)*(1-4*x)^2) + O(x^30)) \\ Colin Barker, Oct 28 2016
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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