cp's OEIS Frontend

A253709 Integer squares c^2 that are equal to the sums of M (A253707) 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).

%I A253709 #33 Sep 08 2022 08:46:10
%S A253709 104329,56205009,3319833924,68869504900,771665618025,5755695204609,
%T A253709 32148582480784,144648440352144,550265331330225,1830621686635225,
%U A253709 5457952678249764,14853496612506084,37420748658691489,88243404864147225,196505988636801600,416206765369428544,843426135281228409,1643334148974958209,3091319880732100900,5634162244739340900
%N A253709 Integer squares c^2 that are equal to the sums of M (A253707) 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 A253709 Numbers a(n)=c^2 such that b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2 has nontrivial solutions over the integers for b being an odd squared integer (A016754) and M (A253707).
%C A253709 To every odd squared integer b (A016754) corresponds a sum of M (A253707) consecutive cubed integers starting at b^3 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) (A253707), c(n)= (b-1)(4b^2-1)/8 = (n(n+1)/2)(4(2n+1)^4-1) (A253708) and a(n)=c(n)^2 (this sequence).
%C A253709 The trivial solutions with M < 1 and b < 2 are not considered here.
%H A253709 Vladimir Pletser, <a href="/A253709/b253709.txt">Table of n, a(n) for n = 1..50000</a>
%H A253709 Vladimir Pletser, <a href="/A253707/a253707_1.txt">File Triplets (M,b,c) for a=(2n+1)^2</a>
%H A253709 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 A253709 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 A253709 <a href="/index/Rec#order_13">Index entries for linear recurrences with constant coefficients</a>, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).
%F A253709 a(n) = ((n(n+1)/2)(4(2n+1)^4-1))^2.
%F A253709 G.f.: -x*(104329*x^10 +54848732*x^9 +2597306469*x^8 +30065816496*x^7 +119309063058*x^6 +186443360232*x^5 +119309063058*x^4 +30065816496*x^3 +2597306469*x^2 +54848732*x +104329) / (x -1)^13. - _Colin Barker_, Jan 10 2015
%e A253709 For n=1, b(1)=9, M(1)=17, c(1)=323, a(1)= 104329 (see File File Triplets (M,b,c) for a=(2n+1)^2 link).
%p A253709 restart: for n from 1 to 50000 do a:=((n*(n+1)/2)(4*(2*n+1)^4-1))^2: print (a); end do:
%t A253709 f[n_] := ((n (n + 1)/2) (4 (2 n + 1)^4 - 1))^2; Array[f, 20] (* _Michael De Vlieger_, Jan 10 2015 *)
%o A253709 (PARI) Vec(-x*(104329*x^10 +54848732*x^9 +2597306469*x^8 +30065816496*x^7 +119309063058*x^6 +186443360232*x^5 +119309063058*x^4 +30065816496*x^3 +2597306469*x^2 +54848732*x +104329) / (x -1)^13 + O(x^100)) \\ _Colin Barker_, Jan 10 2015
%o A253709 (Magma) [((n*(n+1)/2)*(4*(2*n+1)^4-1))^2: n in [1..20]]; // _Vincenzo Librandi_, Jan 15 2015
%Y A253709 Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253680, A253707, A253708.
%K A253709 nonn,easy
%O A253709 1,1
%A A253709 _Vladimir Pletser_, Jan 09 2015