A393694 Numbers that are sums of two positive cubes in two or more ways corresponding to the Pell-type equation X^2 - 3*Y^2 = 6318.

530712, 736097544, 1979009657352, 5347145100161592, 14447982155941656504, 39038442411387919309992, 105481856947214429009618472, 285011938432925772595065675864, 770102152163908417866197413132632, 2080815730134942111139298641754753352

Offset

1,1

Links

Paolo Xausa, Table of n, a(n) for n = 1..250

Jamal Agbanwa, Generating Infinitely Many Taxicab Numbers, Figshare, (preprint) 2025.

Index entries for linear recurrences with constant coefficients, signature (2716,-37830,2716,-1).

Formula

a(n) = x(n-1)^3 + y(n-1)^3 = u(n-1)^3 + w(n-1)^3 where:

x(n) = (-54 + 9*((15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n))/4 + 27,

y(n) = (-162 + 9*((7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n))/4,

u(n) = (-54 + 9*((15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n))/4, and

w(n) = (-162 + 9*((7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n))/4 + 81.

G.f.: 530712*x*(1387 - 38121*x + 2715*x^2 - x^3)/((1 - 2702*x + x^2)*(1 - 14*x + x^2)). - Andrew Howroyd, Feb 25 2026

Example

For n = 8, a(8) = x(7)^3 + y(7)^3 = ((-54 + 9*((15 - 7*sqrt(3))*(7 - 4*sqrt(3))^7 + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^7))/4 + 27)^3 + ((-162 + 9((7 - 5*sqrt(3))*(7 - 4*sqrt(3))^7 + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^7))/4)^3 = 285011938432925772595065675864.

Mathematica

LinearRecurrence[{2716, -37830, 2716, -1}, {530712, 736097544, 1979009657352, 5347145100161592}, 10] (* Paolo Xausa, Mar 12 2026 *)

Crossrefs

Cf. A384106, A389865.

Cf. A011541, A018850.

Subset of A001235.

Sequence in context: A340304 A364903 A210352 * A321828 A038682 A017082

Adjacent sequences: A393691 A393692 A393693 * A393695 A393696 A393697

Keyword

nonn,easy

Author

Jamal Agbanwa, Feb 25 2026

Status

approved