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

Author

Sen-Peng Eu, Apr 05 2001

Keywords

• easy
• nice
• nonn

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).

Crossrefs

Cf. A000330, A007290, A046092, A080838.

Programs

Magma

[n*(2*n-3-(-1)^n)*(2*n-7-(-1)^n)/32 : n in [0..60]]; // Wesley Ivan Hurt, Apr 14 2017

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 *)
LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {0, 0, 0, 0, 0, 5, 6}, 51] (* Mike Sheppard, Feb 17 2025 *)

PARI

a(n)=polcoeff(x^5*(5+x)/(1-x)/(1-x^2)^3+x*O(x^n),n)

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

a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7). - Mike Sheppard, Feb 17 2025