A384106 Numbers representable as the sum of 2 cubes in at least 2 ways generated by a parameterized formula involving (7+4*sqrt(3))^n and (7-4*sqrt(3))^n.

1009736, 2714690888, 7334904115448, 19818905563705976, 53550675461437475048, 144693905277386048024168, 390962878508814502873889816, 1056203940519850679825934312168, 2853755704387709706549646191448888, 7710144396612746633517746345789261976

Offset

1,1

Comments

A rapidly growing sequence of integers, each equal to x(n)^3 + y(n)^3 = u(n)^3 + w(n)^3 for distinct positive integers x(n), y(n), u(n), w(n), generated from a parameterized expression. Values omit small classical examples (like 1729) and begin at much larger values and is therefore a parameterized subset of solutions to A001235.

This sequence also corresponds to the Pell-type equation x^2 - 3*y^2 = 78. - Jamal Agbanwa, May 20 2026

Links

Table of n, a(n) for n=1..10.

Jamal Agbanwa, A Closed-Form Symbolic Generator: A^n + B^n = C^n + D^n, n = 2, 3, Preprint, 2025. See also arXiv:2506.19173 [math.GM], 2025, p. 8.

Formula

a(n) = x(n)^3 + y(n)^3 = u(n)^3 + w(n)^3 where:

x(n) = (-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n)/4 + 3,

y(n) = (-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n)/4,

u(n) = (-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n)/4, abd

w(n) = (-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n)/4 + 9.

Example

For n = 7, a(7) = x(n)^3 + y(n)^3 = ((-6 + (15 - 7*sqrt(3))*(7 - 4*sqrt(3))^7 + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^7)/4 + 3)^3 + ((-18 + (7 - 5*sqrt(3))*(7 - 4*sqrt(3))^7 + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^7)/4)^3 = 390962878508814502873889816.

Crossrefs

Cf. A011541, A018850.

Subset of A001235.

Sequence in context: A330519 A388141 A066354 * A220289 A133219 A306515

Adjacent sequences: A384103 A384104 A384105 * A384107 A384108 A384109

Keyword

nonn,changed

Author

Jamal Agbanwa, May 19 2025

Status

approved