A085409 Sum of three solutions of the Diophantine equation x^2 - y^2 = z^3.

0, 12, 84, 270, 624, 1200, 2052, 3234, 4800, 6804, 9300, 12342, 15984, 20280, 25284, 31050, 37632, 45084, 53460, 62814, 73200, 84672, 97284, 111090, 126144, 142500, 160212, 179334, 199920, 222024, 245700, 271002, 297984, 326700, 357204, 389550, 423792, 459984

Offset: 0

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 13 2003

Keywords

• nonn
• easy

Comments

Parametric representation of the solution is (x, y, z) = (6n^3, 3n^3, 3n^2), thus getting a(n) = 9n^3 + 3n^2.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A085377.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( 6*x*(2 + 6*x + x^2) /(1 - x)^4)); // Marius A. Burtea, Oct 25 2019

Mathematica

Table[9n^3 + 3n^2, {n, 0, 34}]

PARI

concat(0, Vec(6*x*(2 + 6*x + x^2) /(1 - x)^4 + O(x^40))) \\ Colin Barker, Oct 25 2019

Formula

a(n) = 9*n^3 + 3*n^2.

From Colin Barker, Oct 25 2019: (Start)

G.f.: 6*x*(2 + 6*x + x^2) /(1 - x)^4.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>3.

(End)

From Amiram Eldar, Jan 10 2023: (Start)

Sum_{n>=1} 1/a(n) = Pi^2/18 + sqrt(3)*Pi/6 + 3*log(3)/2 - 3.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/36 - Pi/sqrt(3) - 2*log(2) + 3. (End)

Extensions

More terms from Robert G. Wilson v, Aug 16 2003