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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. 3
137, 6341, 291593, 13406981, 616429577, 28342353605, 1303131836297, 59915722116101, 2754820085504393, 126661808211086021, 5823688357624452617, 267763002642513734405, 12311274433198007330057, 566050860924465823448261, 26026028328092229871289993 (list; graph; refs; listen; history; text; internal format)
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 * A103878 A068154 A134874

Adjacent sequences:  A261970 A261971 A261972 * A261974 A261975 A261976

KEYWORD

nonn,easy

AUTHOR

Colin Barker, Sep 07 2015

STATUS

approved

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Last modified September 21 00:17 EDT 2019. Contains 327252 sequences. (Running on oeis4.)