A066456 Upper bound on number of regular triangulations of cyclic polytope C(n, n-4).
1, 1, 2, 4, 8, 14, 25, 40, 65, 97, 146, 206, 292, 394, 533, 694, 905, 1145, 1450, 1792, 2216, 2686, 3257, 3884, 4633, 5449, 6410, 7450, 8660, 9962, 11461, 13066, 14897, 16849, 19058, 21404, 24040, 26830, 29945, 33232, 36881, 40721, 44962, 49414, 54308, 59434
Offset: 1
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- M. Azaola and F. Santos, The number of triangulations of the cyclic polytope C(n,n-4), Discrete Comput. Geom., 27 (2002), 29-48.
- Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).
Programs
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Maple
A066456 := proc(n) local m; if n mod 2 = 0 then m := n/2; 6*binomial(m,4)+3*binomial(m,3)+4*binomial(m,2)-m+2; else m := (n+1)/2; 6*binomial(m,4)+5*binomial(m,2)-4*m+5; fi; end;
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Mathematica
CoefficientList[Series[-(1 - x - 4 x^5 + x^6 - 2 x^2 + 4 x^3 + 2 x^4 + 2 x^7)/((1 + x)^3 (x - 1)^5), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 07 2014 *) -
PARI
Vec(x*(1 - x - 2*x^2 + 4*x^3 + 2*x^4 - 4*x^5 + x^6 + 2*x^7) / ((1 - x)^5*(1 + x)^3) + O(x^60)) \\ Colin Barker, May 04 2017
Formula
G.f.: x*(1-x-4*x^5+x^6-2*x^2+4*x^3+2*x^4+2*x^7) / ( (1+x)^3*(1-x)^5 ). - R. J. Mathar, Aug 07 2014
From Colin Barker, May 04 2017: (Start)
a(n) = (n^4 - 8*n^3 + 52*n^2 - 112*n + 128) / 64 for n even.
a(n) = (n^4 - 8*n^3 + 54*n^2 - 120*n + 137) / 64 for n odd.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n>8.
(End)