A141326 Subsequence of 'Fermat near misses' which is generated by a simple formula based on the cubic binomial expansion along with formulas for the corresponding terms in the expression, x^3 + y^3 = z^3 + 1.

12, 150, 738, 2316, 5640, 11682, 21630, 36888, 59076, 90030, 131802, 186660, 257088, 345786, 455670, 589872, 751740, 944838, 1172946, 1440060, 1750392, 2108370, 2518638, 2986056, 3515700, 4112862, 4783050, 5531988, 6365616, 7290090, 8311782, 9437280, 10673388

Offset: 1

Author

Lewis Mammel (l_mammel(AT)att.net), Aug 03 2008

Keywords

• easy
• nonn

Comments

From Lewis Mammel (l_mammel(AT)att.net), Aug 21 2008: (Start)

In Ramanujan's parametric equation: (ax+y)^3 + (b+x^2y)^3 = (bx+y)^3 + (a+x^2y)^3

where a^2 + ab + b^2 = 3xy^2.

This sequence is obtained by setting a=0, y=1 and finding the solution to b^2=3x:

b=3n, x=3n^2. (End)

Examples

			For a(1)=12: 1 + 12^3 = 9^3 + 10^3 = 1729.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Tito Piezas III and Eric Weisstein, Diophantine Equation--3rd Powers [Lewis Mammel (l_mammel(AT)att.net), Aug 21 2008]
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A050791, A050792, A050793, A050794.

Programs

PARI

Vec(6*x*(2 + 15*x + 18*x^2 + x^3) / (1 - x)^5 + O(x^40)) \\ Colin Barker, Oct 26 2019

Formula

a(n) = 9*n^4 + 3*n, with b(n) = 9*n^4 and c(n) = 9*n^3 + 1 we have 1 + a(n)^3 = b(n)^3 + c(n)^3.

From Colin Barker, Oct 25 2019: (Start)

G.f.: 6*x*(2 + 15*x + 18*x^2 + x^3) / (1 - x)^5.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5.

a(n) = 3*(n + 3*n^4).

(End)

Extensions

Edited by Joerg Arndt, Oct 26 2019