A253707 Numbers M(n) which are the number of terms in the sums of consecutive cubed integers equaling a squared integer, b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2, for a first term b(n) being an odd squared integer (A016754).

17, 98, 291, 644, 1205, 2022, 3143, 4616, 6489, 8810, 11627, 14988, 18941, 23534, 28815, 34832, 41633, 49266, 57779, 67220, 77637, 89078, 101591, 115224, 130025, 146042, 163323, 181916, 201869, 223230, 246047, 270368, 296241, 323714, 352835, 383652, 416213

Offset

1,1

Comments

Numbers M(n) such that b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2 has nontrivial solutions over the integers for b(n) being an odd squared integer (A016754).

To every odd squared integer b corresponds a sum of a consecutive cubed integers starting at b having at least one nontrivial solution. For n>=1, b(n)= (2n+1)^2 (A016754), M(n) = (sqrt(b)-1) (2b-1)/2 = n(8n(n+1)+1) (this sequence), and c(n)= (b-1)(4b^2-1)/8 = (n (n+1)/2)(4(2n+1)^4-1) (A253708).

The trivial solutions with M < 1 and b < 2 are not considered here.

Links

Vladimir Pletser, Table of n, a(n) for n = 1..50000

Vladimir Pletser, File Triplets (M,b,c) for a=(2n+1)^2

Vladimir Pletser, Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares, Research Gate, 2015.

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 (4,-6,4,-1).

Formula

a(n) = n(8n(n+1)+1).

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - Colin Barker, Jan 10 2015

G.f.: x*(x^2+30*x+17) / (x-1)^4. - Colin Barker, Jan 10 2015

Example

For n=1, b(n)=9, M(n)=17, c(n)=323 (see File Triplets link).

Maple

restart: for n from 1 to 50000 do a:= n*(8*n*(n+1)+1): print (a); end do:

Mathematica

f[n_] := n*(8 n (n + 1) + 1); Array[f, 52] (* Michael De Vlieger, Jan 10 2015 *)
LinearRecurrence[{4, -6, 4, -1}, {17, 98, 291, 644}, 40] (* Harvey P. Dale, Jul 31 2018 *)

Prog

(PARI) Vec(x*(x^2+30*x+17)/(x-1)^4 + O(x^100)) \\ Colin Barker, Jan 10 2015
(Magma) [n*(8*n*(n+1)+1): n in [1..40]]; // Vincenzo Librandi, Feb 19 2015

Crossrefs

Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253680, A253681, A253708, A253709.

Sequence in context: A231667 A044268 A044649 * A160827 A231671 A231675

Adjacent sequences: A253704 A253705 A253706 * A253708 A253709 A253710

Keyword

nonn,easy

Author

Vladimir Pletser, Jan 09 2015

Status

approved