A060423 Number of obtuse triangles made from vertices of a regular n-gon.

0, 0, 0, 0, 0, 5, 6, 21, 24, 54, 60, 110, 120, 195, 210, 315, 336, 476, 504, 684, 720, 945, 990, 1265, 1320, 1650, 1716, 2106, 2184, 2639, 2730, 3255, 3360, 3960, 4080, 4760, 4896, 5661, 5814, 6669, 6840, 7790, 7980, 9030, 9240, 10395, 10626

Offset

0,6

Links

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Léo Ducas, Kissing Number of Craig's Lattice and Spherical Decoding, Bachelor's seminar AGM Spring 2025, Leiden Univ. (Netherlands, 2024). See p. 2.

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

Formula

a(n) = n*(n-1)*(n-3)/8 when n odd; n*(n-2)*(n-4)/8 when n even.

G.f.: x^5*(x+5)/((1-x)(1-x^2)^3). - Michael Somos, Jan 30 2004

For n odd, a(n) = A080838(n). - Gerald McGarvey, Sep 14 2008

a(n) = n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32. - Wesley Ivan Hurt, Dec 31 2013

E.g.f.: x*((x - 3)*x*cosh(x) + (x^2 - x + 3)*sinh(x))/8. - Stefano Spezia, May 28 2022

Maple

A060423:=n->n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32; seq(A060423(n), n=0..100); # Wesley Ivan Hurt, Dec 31 2013

Mathematica

Table[n(2n-3-(-1)^n)(2n-7-(-1)^n)/32, {n, 0, 100}] (* Wesley Ivan Hurt, Dec 31 2013 *)
Table[If[EvenQ[n], (n(n-2)(n-4))/8, (n(n-1)(n-3))/8], {n, 0, 50}] (* Harvey P. Dale, Sep 18 2018 *)

Prog

(PARI) a(n)=polcoeff(x^5*(5+x)/(1-x)/(1-x^2)^3+x*O(x^n), n)
(Magma) [n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32 : n in [0..60]]; // Wesley Ivan Hurt, Apr 14 2017

Crossrefs

Cf. A000330, A007290, A046092, A080838.

Sequence in context: A351950 A327860 A057520 * A037951 A095308 A132796

Adjacent sequences: A060420 A060421 A060422 * A060424 A060425 A060426

Keyword

easy,nice,nonn,changed

Author

Sen-Peng Eu, Apr 05 2001

Status

approved