A395378 Numbers that are sums of two positive cubes in two or more ways corresponding to the Pell-type equation X^2 - 3*Y^2 = 56862.

14329224, 19874633688, 53433260748504, 144372917704362984, 390095518210424725608, 1054037945107473821369784, 2848010137574789583259698744, 7695322337688995860066773248328, 20792758108425527282387330154581064, 56182024713643437000761063327378340504

Offset

1,1

Links

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

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

Formula

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

x(n) = (-162 + 27*((15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n))/4 + 81,

y(n) = (-486 + 27*((7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n))/4,

u(n) = (-162 + 27*((15 - 7*sqrt(3))*(7 - 4*sqrt(3))^n + (15 + 7*sqrt(3))*(7 + 4*sqrt(3))^n))/4, and

w(n) = (-486 + 27*((7 - 5*sqrt(3))*(7 - 4*sqrt(3))^n + (7 + 5*sqrt(3))*(7 + 4*sqrt(3))^n))/4 + 243.

Example

For n = 9, a(9) = x(8)^3 + y(8)^3 = 259312805379^3 + 149714317845^3 = 20792758108425527282387330154581064.

Mathematica

LinearRecurrence[{2716, -37830, 2716, -1}, {14329224, 19874633688, 53433260748504, 144372917704362984}, 10]

Crossrefs

Cf. A384106, A389865.

Cf. A393694, A018850.

Subset of A001235.

Sequence in context: A395550 A204952 A210019 * A111346 A238080 A251009

Adjacent sequences: A395375 A395376 A395377 * A395379 A395380 A395381

Keyword

nonn,easy

Author

Jamal Agbanwa, Apr 20 2026

Status

approved