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Numbers of isomers of unbranched a-4-catapolypentagons - see Brunvoll reference for precise definition.
2

%I #24 Mar 31 2020 12:41:38

%S 1,3,13,45,164,527,1652,4876,14000,38800,105112,278104,722736,1846320,

%T 4650368,11560320,28413312,69111168,166551680,397975680,943672576,

%U 2221821184,5197313024,12084884480,27944656896,64286203904,147182565376

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

%H 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.

%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (10,-34,20,148,-328,8,688,-640,-256,640,-256).

%F G.f.: x^4 * (64*x^13 -32*x^12 -192*x^11 +128*x^10 +212*x^9 -172*x^8 -114*x^7 +110*x^6 +41*x^5 -52*x^4 -3*x^3 +17*x^2 -7*x+1) / ((2*x-1)^5 * (2*x^2-1)^3). - _Colin Barker_, Nov 30 2012

%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: A121136 := proc(n) Ifunc(n,4,5) ; end: for n from 4 to 80 do printf("%d,",A121136(n)) ; od: # _R. J. Mathar_, Aug 07 2008

%t Join[{1, 3, 13}, LinearRecurrence[{10, -34, 20, 148, -328, 8, 688, -640, -256, 640, -256}, {45, 164, 527, 1652, 4876, 14000, 38800, 105112, 278104, 722736, 1846320}, 24]] (* _Jean-François Alcover_, Mar 31 2020 *)

%Y Cf. A121133. - _R. J. Mathar_, Aug 07 2008

%K nonn,easy

%O 4,2

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

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