A024163 Number of integer-sided triangles with sides a,b,c, a

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 3, 1, 3, 3, 3, 3, 6, 3, 6, 6, 6, 6, 10, 6, 10, 10, 10, 10, 15, 10, 15, 15, 15, 15, 21, 15, 21, 21, 21, 21, 28, 21, 28, 28, 28, 28, 36, 28, 36, 36, 36, 36, 45, 36, 45, 45, 45, 45, 55, 45, 55, 55, 55, 55, 66, 55, 66, 66, 66, 66, 78, 66, 78, 78, 78, 78, 91

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

Number of partitions of n - 11 into parts 2, 3, and 6. - Stefano Spezia, Aug 21 2025

Examples

			2,4,5 for n=11 is the smallest such triangle.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,-1,1,0,-1,-1,0,1).

Crossrefs

Cf. A024165, A025796.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 100);
[0,0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^11/((1-x^2)*(1-x^3)*(1-x^6)) )); // G. C. Greubel, Jul 03 2021

Mathematica

CoefficientList[Series[x^10/((1-x^2)(1-x^3)(1-x^6)), {x, 0, 100}], x] (* Vincenzo Librandi, Oct 28 2014 *)

PARI

concat(vector(11,i,0), Vec(1/(1-x^2)/(1-x^3)/(1-x^6)+O(x^99))) \\ Charles R Greathouse IV, Oct 28 2014

Sage

def A024163_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( x^11/((1-x^2)*(1-x^3)*(1-x^6)) ).list()
a=A024163_list(100); a[1:] # G. C. Greubel, Jul 03 2021

Formula

G.f.: x^11/((1-x^2)*(1-x^3)*(1-x^6)). - Tani Akinari, Oct 27 2014