A163827 a(n) = 6n^3 + 1, solution z in Diophantine equation x^3 + y^3 = z^3 - 2. It may be considered a Fermat near miss by 2.

7, 49, 163, 385, 751, 1297, 2059, 3073, 4375, 6001, 7987, 10369, 13183, 16465, 20251, 24577, 29479, 34993, 41155, 48001, 55567, 63889, 73003, 82945, 93751, 105457, 118099, 131713, 146335, 162001, 178747, 196609, 215623, 235825, 257251, 279937

Offset

1,1

Comments

It is easy to check that with x = 6n^2, y = 6n^3 - 1, and this z = 6n^3 + 1, it satisfies the Diophantine equation x^3 + y^3 = z^3 - 2. Thus these are near-misses for Fermat equation.

For n>2, it seems to be the only solution of x^n + y^n = z^n - 2 (or even that differ by 2 from FLT, see A050787 and A050791 for solutions that differ by 1). As 2 is not a cube, these solutions are not included in the theory for x^3 + y^3 = u^3 + v^3.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

Formula

a(n) = 6n^3+1.

a(1)=7, a(2)=49, a(3)=163, a(4)=385, a(n)=4*a(n-1)-6*a(n-2)+4*a(n-3)- a(n-4). [Harvey P. Dale, Dec 12 2011]

G.f.: (-x^3+9*x^2+21*x+7)/(x-1)^4. [Harvey P. Dale, Dec 12 2011]

Example

For n=1, a(1)=7 and 7^3 - 2 (=341) = 5^3 + 6^3.
For n=2, a(2)=49 and 49^3 - 2 (=117647) = 24^3 + 47^3.

Maple

A163827:=n->6*n^3+1: seq(A163827(n), n=1..50); # Wesley Ivan Hurt, Jan 09 2017

Mathematica

6*Range[40]^3+1 (* or *) LinearRecurrence[{4, -6, 4, -1}, {7, 49, 163, 385}, 40] (* Harvey P. Dale, Dec 12 2011 *)

Prog

(PARI) a(n)=6*n^3+1 \\ Charles R Greathouse IV, Nov 29 2014
(Magma) [6*n^3+1 : n in [1..50]]; // Wesley Ivan Hurt, Jan 09 2017

Crossrefs

Cf. A050787, A050791.

Sequence in context: A152777 A003530 A262269 * A206989 A221962 A373684

Adjacent sequences: A163824 A163825 A163826 * A163828 A163829 A163830

Keyword

nonn,easy

Author

Carlos Alves, Aug 05 2009

Status

approved