A262142 The first of ten consecutive positive integers the sum of the squares of which is equal to the sum of the squares of nine consecutive positive integers.

171, 6660, 253071, 9610200, 364934691, 13857908220, 526235577831, 19983094049520, 758831338304091, 28815607761506100, 1094234263598927871, 41552086408997753160, 1577885049278315692371, 59918079786166998557100, 2275309146825067629477591

Offset

1,1

Comments

For the first of the corresponding nine consecutive positive integers, see A262141.

Links

Colin Barker, Table of n, a(n) for n = 1..632

Index entries for linear recurrences with constant coefficients, signature (39,-39,1).

Formula

a(n) = 39*a(n-1)-39*a(n-2)+a(n-3) for n>3.

G.f.: 9*x*(x-19) / ((x-1)*(x^2-38*x+1)).

a(n) = 3*(-6-(-3+sqrt(10))*(19+6*sqrt(10))^(-n)+(3+sqrt(10))*(19+6*sqrt(10))^n)/4. - Colin Barker, Mar 03 2016

Example

171 is in the sequence because 171^2 + ... + 180^2 = 308085 = 181^2 + ... + 189^2.

Mathematica

LinearRecurrence[{39, -39, 1}, {171, 6660, 253071}, 30] (* Harvey P. Dale, Sep 26 2015 *)

Prog

(PARI) Vec(9*x*(x-19)/((x-1)*(x^2-38*x+1)) + O(x^30))

Crossrefs

Cf. A262141.

Sequence in context: A332418 A152926 A239270 * A206595 A127959 A046166

Adjacent sequences: A262139 A262140 A262141 * A262143 A262144 A262145

Keyword

nonn,easy

Author

Colin Barker, Sep 12 2015

Status

approved