A262075 The first of eight consecutive positive integers the sum of the squares of which is equal to the sum of the squares of seven consecutive positive integers.

105, 3248, 97433, 2919840, 87497865, 2622016208, 78572988473, 2354567638080, 70558456154025, 2114399116982768, 63361415053329113, 1898728052482890720, 56898480159433392585, 1705055676730518886928, 51094771821756133215353, 1531138098975953477573760

Offset

1,1

Comments

For the first of the corresponding seven consecutive positive integers, see A262074.

Links

Colin Barker, Table of n, a(n) for n = 1..676

Index entries for linear recurrences with constant coefficients, signature (31,-31,1).

Formula

a(n) = 31*a(n-1)-31*a(n-2)+a(n-3) for n>3.

G.f.: 7*x*(x-15) / ((x-1)*(x^2-30*x+1)).

Example

105 is in the sequence because 105^2 + ... + 112^2 (8 terms) = 94220 = 113^2 + ... + 119^2 (7 terms).

Mathematica

LinearRecurrence[{31, -31, 1}, {105, 3248, 97433}, 20] (* Vincenzo Librandi, Sep 11 2015 *)
Table[1/4 (-14-(15-4 Sqrt[14])^n (7+2 Sqrt[14])+(-7+2 Sqrt[14]) (15+4 Sqrt[ 14])^n), {n, 2, 20}]//Simplify (* Harvey P. Dale, Aug 01 2019 *)

Prog

(PARI) Vec(7*x*(x-15)/((x-1)*(x^2-30*x+1)) + O(x^20))
(Magma) I:=[105, 3248, 97433]; [n le 3 select I[n] else 31*Self(n-1)-31*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Sep 11 2015

Crossrefs

Cf. A262074, A262076, A262077.

Sequence in context: A220822 A166821 A166803 * A018233 A154633 A122814

Adjacent sequences: A262072 A262073 A262074 * A262076 A262077 A262078

Keyword

nonn,easy

Author

Colin Barker, Sep 10 2015

Status

approved