A261973 The first of three consecutive positive integers the sum of the squares of which is equal to the sum of the squares of eleven consecutive positive integers.

137, 6341, 291593, 13406981, 616429577, 28342353605, 1303131836297, 59915722116101, 2754820085504393, 126661808211086021, 5823688357624452617, 267763002642513734405, 12311274433198007330057, 566050860924465823448261, 26026028328092229871289993

Offset

1,1

Comments

For the first of the corresponding eleven consecutive positive integers, see A261974.

From Zak Seidov, Sep 07 2015: (Start)

Positive values x of solutions (x, y) to the Diophantine equation 380 + 110x + 11x^2 - 6y - 3y^2 = 0, with values of y in A261974.

Note that there are also solutions with negative x: (x,y) = (-77,137), (-3317, 6341), (-152285, 291593), (-7001573, 13406981), ... with values of y in A261974. (End)

Links

Colin Barker, Table of n, a(n) for n = 1..601

Index entries for linear recurrences with constant coefficients, signature (47,-47,1).

Formula

a(n) = 47*a(n-1)-47*a(n-2)+a(n-3) for n>3.

G.f.: -x*(5*x^2-98*x+137) / ((x-1)*(x^2-46*x+1)).

a(n) = -1+3*(23+4*sqrt(33))^(-n)+3*(23+4*sqrt(33))^n. - Colin Barker, Mar 03 2016

Example

137 is in the sequence because 137^2 + 138^2 + 139^2 = 57134 = 67^2 + ... + 77^2.

Mathematica

LinearRecurrence[{47, -47, 1}, {137, 6341, 291593}, 20] (* Vincenzo Librandi, Sep 08 2015 *)

Prog

(PARI) Vec(-x*(5*x^2-98*x+137) / ((x-1)*(x^2-46*x+1)) + O(x^40))
(Magma) I:=[137, 6341, 291593]; [n le 3 select I[n] else 47*Self(n-1)-47*Self(n-2)+Self(n-3): n in [1..15]]; // Vincenzo Librandi, Sep 08 2015

Crossrefs

Cf. A157088, A261972, A261974.

Sequence in context: A221346 A138348 A278175 * A357131 A103878 A068154

Adjacent sequences: A261970 A261971 A261972 * A261974 A261975 A261976

Keyword

nonn,easy

Author

Colin Barker, Sep 07 2015

Status

approved