A373561 a(n) = (1/3)*n^5 + (1/2)*n^4 + (1/6)*n^3.

0, 1, 20, 126, 480, 1375, 3276, 6860, 13056, 23085, 38500, 61226, 93600, 138411, 198940, 279000, 382976, 515865, 683316, 891670, 1148000, 1460151, 1836780, 2287396, 2822400, 3453125, 4191876, 5051970, 6047776, 7194755, 8509500, 10009776, 11714560, 13644081, 15819860

Offset: 0

Author

Mats Granvik, Jun 10 2024

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Crossrefs

Cf. A368197, A000330.

Programs

Mathematica

nn = 34; Table[+1/3  n^5 + 1/2  n^4 + 1/6  n^3, {n, 0, nn}]
p = 2; Table[Sum[Sum[Sum[Sum[If[GCD[x^p + y^p - z^p, n] == k, x^p + y^p - z^p, 0], {x, 1, n}], {y, 1, n}], {z, 1, n}], {k, 1, n}], {n, 0, nn}]
LinearRecurrence[{6,-15,20,-15,6,-1}, {0,1,20,126,480,1375}, 35] (* Hugo Pfoertner, Jun 10 2024 *)

Formula

a(n) = n^3*(n+1)*(2*n+1)/6.

a(n) = n^2 * A000330(n).

Conjecture: a(n) = Sum_{k=1..n} Sum_{z=1..n} Sum_{y=1..n} Sum_{x=1..n} [GCD(f(x,y,z), n) = k] * f(x,y,z), where f(x,y,z) = x^2 + y^2 - z^2.

G.f.: x*(1 + 14*x + 21*x^2 + 4*x^3)/(1 - x)^6. - Stefano Spezia, Jun 10 2024