A085377 a(n) = 15n^2 + 13n^3.

0, 28, 164, 486, 1072, 2000, 3348, 5194, 7616, 10692, 14500, 19118, 24624, 31096, 38612, 47250, 57088, 68204, 80676, 94582, 110000, 127008, 145684, 166106, 188352, 212500, 238628, 266814, 297136, 329672, 364500, 401698, 441344, 483516, 528292

Offset: 0

Author

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Aug 12 2003

Keywords

• nonn

Comments

Numbers that are the sum of three solutions of the Diophantine equation x^3 - y^3 = z^2.

Parametric representation of the solution is (x,y,z) = (8n^2, 7n^2, 13n^3), thus getting a(n) = 8n^2 + 7n^2 + 13n^3 = 15n^2 + 13n^3.

Geometrically, 13^2 = 8^3 - 7^3 means that the square of the hypotenuse of a Pythagorean triangle (5,12,13) is the difference of two cubes, which I recently found on p70 of David Wells' book "The Penguin Dictionary of Curios and Interesting Numbers", Penguin Books, 1997. See also A085479.

Crossrefs

Cf. A085409.

Programs

Mathematica

Table[15n^2 + 13n^3, {n, 1, 34}]

Formula

a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). G.f.: 2*x*(14+26*x-x^2)/(1-x)^4. [From R. J. Mathar, Apr 20 2009]

Extensions

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

Edited by N. J. A. Sloane, Apr 29 2008