A066456 - OEIS

%I #18 May 04 2017 08:24:48

%S 1,1,2,4,8,14,25,40,65,97,146,206,292,394,533,694,905,1145,1450,1792,

%T 2216,2686,3257,3884,4633,5449,6410,7450,8660,9962,11461,13066,14897,

%U 16849,19058,21404,24040,26830,29945,33232,36881,40721,44962,49414,54308,59434

%N Upper bound on number of regular triangulations of cyclic polytope C(n, n-4).

%H Vincenzo Librandi, <a href="/A066456/b066456.txt">Table of n, a(n) for n = 1..1000</a>

%H M. Azaola and F. Santos, <a href="http://personales.unican.es/santosf/Articulos/">The number of triangulations of the cyclic polytope C(n,n-4)</a>, Discrete Comput. Geom., 27 (2002), 29-48.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-6,0,6,-2,-2,1).

%F 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

%F From _Colin Barker_, May 04 2017: (Start)

%F a(n) = (n^4 - 8*n^3 + 52*n^2 - 112*n + 128) / 64 for n even.

%F a(n) = (n^4 - 8*n^3 + 54*n^2 - 120*n + 137) / 64 for n odd.

%F 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.

%F (End)

%p 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;

%t 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 *)

%o (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

%Y Cf. A066375 (bisection), A066455 (bisection).

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, Jan 04 2002