%I #22 Jul 07 2024 18:02:16

%S 1,1,3,10,46,192,840,3584,15392,65536,278656,1179648,4981248,20971520,

%T 88082432,369098752,1543512064,6442450944,26843578368,111669149696,

%U 463856599040,1924145348608,7971459825664,32985348833280,136339443941376,562949953421312

%N Numbers of isomers of unbranched a-4-catapolyheptagons - see Brunvoll reference for precise definition.

%D J. Brunvoll, S. J. Cyvin and B. N. Cyvin, Isomer enumeration of polygonal systems..., J. Molec. Struct. (Theochem), 364 (1996), 1-13.

%H Vincenzo Librandi, <a href="/A121138/b121138.txt">Table of n, a(n) for n = 1..300</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-12,-32,64).

%F Empirical (for n>=4): a(n) = 2^(n-8)*((n+6)*2^n + 8 - 8*(-1)^n). - _Vaclav Kotesovec_, Nov 29 2012

%F 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

%F From _Colin Barker_, Oct 28 2016: (Start)

%F a(n) = 4^(n-4)*(n+6) for n>3 and even.

%F a(n) = 2^(n-8)*(2^n*(n+6)+16) for n>3 and odd.

%F a(n) = 8*a(n-1)-12*a(n-2)-32*a(n-3)+64*a(n-4) for n>7.

%F (End)

%p 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

%t 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 *)

%t LinearRecurrence[{8,-12,-32,64},{1,1,3,10,46,192,840},30] (* _Harvey P. Dale_, Jul 07 2024 *)

%o (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

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, Aug 13 2006

%E Extended beyond a(10) by _R. J. Mathar_, Aug 07 2008