A262063 The first of seven consecutive positive integers the sum of the squares of which is equal to the sum of the squares of six consecutive positive integers.

78, 2100, 54594, 1417416, 36798294, 955338300, 24801997578, 643896598800, 16716509571294, 433985352254916, 11266902649056594, 292505483523216600, 7593875668954575078, 197148261909295735500, 5118260933972734547994, 132877636021381802512416

Offset

1,1

Comments

For the first of the corresponding six consecutive positive integers, see A262062.

Links

Colin Barker, Table of n, a(n) for n = 1..706

Index entries for linear recurrences with constant coefficients, signature (27,-27,1).

Formula

a(n) = 27*a(n-1)-27*a(n-2)+a(n-3) for n>3.

G.f.: 6*x*(x-13) / ((x-1)*(x^2-26*x+1)).

a(n) = (13+2*sqrt(42))^(-n)*(6-sqrt(42)+(6+sqrt(42))*(13+2*sqrt(42))^(2*n))/4-3. - Colin Barker, Mar 03 2016

Example

78 is in the sequence because 78^2 + ... + 84^2 = 45955 = 85^2 + ... + 90^2.

Mathematica

CoefficientList[Series[6 (x - 13)/((x - 1) (x^2 - 26 x + 1)), {x, 0, 20}], x] (* Vincenzo Librandi, Sep 10 2015 *)
LinearRecurrence[{27, -27, 1}, {78, 2100, 54594}, 30] (* Harvey P. Dale, May 17 2018 *)

Prog

(PARI) Vec(6*x*(x-13)/((x-1)*(x^2-26*x+1)) + O(x^20))
(Magma) I:=[78, 2100, 54594]; [n le 3 select I[n] else 27*Self(n-1)-27*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Sep 10 2015

Crossrefs

Cf. A262062.

Sequence in context: A187590 A172217 A036524 * A140934 A133239 A210407

Adjacent sequences: A262060 A262061 A262062 * A262064 A262065 A262066

Keyword

nonn,easy

Author

Colin Barker, Sep 09 2015

Status

approved