A253707 - OEIS

%I #45 Sep 08 2022 08:46:10

%S 17,98,291,644,1205,2022,3143,4616,6489,8810,11627,14988,18941,23534,

%T 28815,34832,41633,49266,57779,67220,77637,89078,101591,115224,130025,

%U 146042,163323,181916,201869,223230,246047,270368,296241,323714,352835,383652,416213

%N 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).

%C 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).

%C 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).

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

%H Vladimir Pletser, <a href="/A253707/b253707.txt">Table of n, a(n) for n = 1..50000</a>

%H Vladimir Pletser, <a href="/A253707/a253707_1.txt">File Triplets (M,b,c) for a=(2n+1)^2</a>

%H Vladimir Pletser, <a href="http://www.researchgate.net/profile/Vladimir_Pletser/publication/271272786">Number of terms, first term and square root of sums of consecutive cubed integers equal to integer squares</a>, Research Gate, 2015.

%H Vladimir Pletser, <a href="http://arxiv.org/abs/1501.06098">General solutions of sums of consecutive cubed integers equal to squared integers</a>, arXiv:1501.06098 [math.NT], 2015.

%H R. J. Stroeker, <a href="http://www.numdam.org/item?id=CM_1995__97_1-2_295_0">On the sum of consecutive cubes being a perfect square</a>, Compositio Mathematica, 97 no. 1-2 (1995), pp. 295-307.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

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

%F a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)-a(n-4). - _Colin Barker_, Jan 10 2015

%F G.f.: x*(x^2+30*x+17) / (x-1)^4. - _Colin Barker_, Jan 10 2015

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

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

%t f[n_] := n*(8 n (n + 1) + 1); Array[f, 52] (* _Michael De Vlieger_, Jan 10 2015 *)

%t LinearRecurrence[{4,-6,4,-1},{17,98,291,644},40] (* _Harvey P. Dale_, Jul 31 2018 *)

%o (PARI) Vec(x*(x^2+30*x+17)/(x-1)^4 + O(x^100)) \\ _Colin Barker_, Jan 10 2015

%o (Magma) [n*(8*n*(n+1)+1): n in [1..40]]; // _Vincenzo Librandi_, Feb 19 2015

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

%K nonn,easy

%O 1,1

%A _Vladimir Pletser_, Jan 09 2015