A066455 -id:A066455 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 04 2002

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).

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

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;

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 *)

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

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)

A112830 Table of number of domino tilings of generalized Aztec pillows of type (1, ..., 1, 3, 1, ..., 1)_n.

Original entry on oeis.org

1, 1, 5, 1, 10, 25, 1, 17, 65, 113, 1, 26, 146, 346, 481, 1, 37, 292, 932, 1637, 1985, 1, 50, 533, 2248, 5013, 7218, 8065, 1, 65, 905, 4937, 13897, 24201, 30529, 32513, 1, 82, 1450, 10018, 35218, 74530, 108970, 126034, 130561, 1, 101, 2216, 19016, 82436

Offset: 0

Christopher Hanusa (chanusa(AT)math.binghamton.edu), Sep 21 2005

The number of tilings of a generalized Aztec pillow of type (k 1's followed by a 3 followed by n-k-1 1's) is entry (n,k+1).

			The number of tilings of a generalized Aztec pillow of type (1,1,3,1)_n is entry (4,3) = 346.

C. Hanusa, A Gessel-Viennot-Type Method for Cycle Systems with Applications to Aztec Pillows, PhD Thesis, 2005, University of Washington, Seattle, USA.

A092440 (main diagonal), A092441 (first subdiagonal), A002522 (column k = 1), A066455 (column k = 2). Cf. A264960.

matrix(11,11,[seq([seq(((2^n-sum(binomial(n,j),j=0..k))^2+(binomial(n-1,k))^2)/2,n=k+1..k+11)],k=0..10)]);

T(2*n,n) = A264960(n). - Peter Bala, Nov 29 2015

cp's OEIS Frontend

A066456 Upper bound on number of regular triangulations of cyclic polytope C(n, n-4).

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