A024165 Number of integer-sided triangles with sides a,b,c, a b - a.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 2, 1, 1, 2, 2, 1, 4, 2, 2, 4, 4, 2, 6, 4, 4, 6, 6, 4, 9, 6, 6, 9, 9, 6, 12, 9, 9, 12, 12, 9, 16, 12, 12, 16, 16, 12, 20, 16, 16, 20, 20, 16, 25, 20, 20, 25, 25, 20, 30, 25, 25, 30, 30, 25, 36, 30, 30, 36, 36, 30, 42, 36, 36, 42, 42, 36, 49, 42, 42, 49, 49

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy
• changed

Comments

Same as A025828 with zeros prepended. - Joerg Arndt, Nov 04 2014

Links

• G. C. Greubel, Table of n, a(n) for n = 1..999
• Index entries for linear recurrences with constant coefficients, signature (0,0,1,1,0,1,-1,0,-1,-1,0,0,1).

Crossrefs

Cf. A024163, A024164, A025828.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 100);
[0,0,0,0,0,0,0,0,0,0,0,0] cat Coefficients(R!( x^13/((1-x^3)*(1-x^4)*(1-x^6)) )); // G. C. Greubel, Jul 03 2021

Mathematica

LinearRecurrence[{0,0,1,1,0,1,-1,0,-1,-1,0,0,1},{0,0,0,0,0,0,0,0,0,0,0,0,1},100] (* Harvey P. Dale, Sep 04 2017 *)

PARI

a(n) = ((n-1)\3 - n\4)*((n-1)\3 + n\4 - n\2) \\ Hoang Xuan Thanh, Aug 31 2025

Sage

def A024165_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( x^13/((1-x^3)*(1-x^4)*(1-x^6)) ).list()
a=A024165_list(100); a[1:] # G. C. Greubel, Jul 03 2021

Formula

G.f.: x^13/((1-x^3)*(1-x^4)*(1-x^6)). - Tani Akinari, Nov 04 2014

From Robert Israel, Nov 04 2014: (Start)

a(n) = a(n-3) + a(n-4) + a(n-6) - a(n-7) - a(n-9) - a(n-10) + a(n-13) for n >= 14.

a(6*n) = (2*n^2 - 8*n + 7 + (-1)^n)/8, n >= 1.

a(6*n+1) = a(6*n+4) = a(6*n+5) = (2*n^2 - 1 + (-1)^n)/8.

a(6*n+2) = a(6*n+3) = (2*n^2 - 4*n + 1 - (-1)^n)/8. (End)

From Hoang Xuan Thanh, Aug 31 2025: (Start)

a(n) = floor((n^2 -5*n +40 -(n-13)*(3*(-1)^n +8*((n+2) mod 3)) -12*((n+5) mod 6))/144).

a(n) = (floor((n-1)/3) - floor(n/4))*(floor((n-1)/3) + floor(n/4) - floor(n/2)). (End)