A180920 Numbers k such that the sum of the cubes of the k consecutive integers starting from k is a square.

1, 33, 2017, 124993, 7747521, 480221281, 29765971873, 1845010034817, 114360856186753, 7088528073543841, 439374379703531361, 27234123013545400513, 1688076252460111300417, 104633493529513355225313, 6485588522577367912668961, 402001854906267297230250241

Offset

1,2

Comments

Numbers k such that A240137(k) is a square.

A240137(k) is a square iff (3n-1)*(5n-3) is a square. 15*n^2-14*n+3 = k^2 can be solved as a Pell equation resulting in formula below confirming linear recurrence from Colin Barker. - Ray Chandler, Jan 12 2024

Links

Colin Barker, Table of n, a(n) for n = 1..558

Vladimir Pletser, General solutions of sums of consecutive cubed integers equal to squared integers, arXiv:1501.06098 [math.NT], 2015.

R. J. Stroeker, On the sum of consecutive cubes being a perfect square, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.

Index entries for linear recurrences with constant coefficients, signature (63,-63,1).

Formula

a(n) = 31*a(n-1) - 14 + 8*sqrt((3*a(n-1) - 1)*(5*a(n-1) - 3)).

From Colin Barker, Feb 18 2015: (Start)

a(n) = 63*a(n-1) - 63*a(n-2) + a(n-3).

G.f.: -x*(x^2-30*x+1) / ((x-1)*(x^2-62*x+1)). (End)

a(n) = (14+2*(4*(31-8*Sqrt(15))^n+Sqrt(15)*(31-8*Sqrt(15))^n+4*(31+8*Sqrt(15))^n-Sqrt(15)*(31+8*Sqrt(15))^n))/30. - Ray Chandler, Jan 12 2024

Mathematica

a[1] = 1; a[n_] := a[n] = 31 a[n - 1] - 14 + 8 Sqrt[(3 a[n - 1] - 1)*(5 a[n - 1] - 3)]; Array[a, 14] (* Robert G. Wilson v, Sep 27 2010 *)

Prog

(PARI)
default(realprecision, 1000)
vector(20, n, if(n==1, t=1, t=round(31*t-14+8*((3*t-1)*(5*t-3))^(1/2)))) \\ Colin Barker, Feb 19 2015

Crossrefs

Cf. A180921, A240137.

Sequence in context: A330478 A284221 A264170 * A120288 A284112 A099370

Adjacent sequences: A180917 A180918 A180919 * A180921 A180922 A180923

Keyword

nonn,easy

Author

Vladimir Pletser, Sep 24 2010

Extensions

a(8) onwards from Robert G. Wilson v, Sep 27 2010

Name clarified by Jon E. Schoenfield, Mar 11 2022

Status

approved