A253725 - OEIS

%I #34 Feb 18 2023 13:57:38

%S 254016,76055841,4286582784,97578140625,1253342942784,10849309481241,

%T 70360154505216,366000785535609,1599920001000000,6075789839706609,

%U 20542200096854016,62996919308080641,177789795179217984,467082203214515625,1152912708530601984

%N Integer squares c^2 that are equal to the sums of a number M(n) of consecutive cubed integers b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2, starting at b(n) (A002593) for M(n) being twice a squared integer (A001105).

%C Numbers a(n)=c^2 such that b^3 + (b+1)^3 + ... + (b+M-1)^3 = c^2 has nontrivial solutions over the integers where M(n) is twice a squared integer (A001105) and b(n)=(A002593).

%C If M is twice a squared integer, there always exists at least one nontrivial solution for the sum of M consecutive cubed integers starting at b^3 and equaling a squared integer c^2. For n>=1, M(n)= 2n^2 (A001105), b(n) = M(M-1)/2 = n^2(2n^2 - 1) (A002593), c(n)= sqrt(M/2) (M(M^2-1)/2)= n^3(4n^4 - 1) (A253724) and a(n)=c(n)^2 (this sequence).

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

%H Vladimir Pletser, <a href="/A253725/b253725.txt">Table of n, a(n) for n = 2..50000</a>

%H Vladimir Pletser, <a href="/A253724/a253724_1.txt">File Triplets (M,b,c) for M=2n^2</a>

%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_15">Index entries for linear recurrences with constant coefficients</a>, signature (15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1).

%F a(n) = (n^3(4n^4 - 1))^2.

%F G.f.: -9*x^2*(x^14 -15*x^13 +106*x^12 +27754*x^11 +8028759*x^10 +352487303*x^9 +4572193580*x^8 +22833696108*x^7 +49725383807*x^6 +49725372367*x^5 +22833705546*x^4 +4572187210*x^3 +352490761*x^2 +8027289*x +28224) / (x -1)^15. - _Colin Barker_, Jan 14 2015

%e For n=2, M(n)=8, b(n)=28, c(n)=504, a(n)=c^2=254016.

%e See "File Triplets (M,b,c) for M=2n^2" link.

%p restart: for n from 2 to 50000 do a:=(n^3*(4*n^4 - 1))^2: print (a); end do:

%t f[n_] := (n^3 (4 n^4 - 1))^2; Rest[f /@ Range@16] (* _Michael De Vlieger_, Jan 28 2015 *)

%t LinearRecurrence[{15,-105,455,-1365,3003,-5005,6435,-6435,5005,-3003,1365,-455,105,-15,1},{254016,76055841,4286582784,97578140625,1253342942784,10849309481241,70360154505216,366000785535609,1599920001000000,6075789839706609,20542200096854016,62996919308080641,177789795179217984,467082203214515625,1152912708530601984},20] (* _Harvey P. Dale_, Feb 18 2023 *)

%o (Magma) [(n^3*(4*n^4-1))^2: n in [2..20]]; // _Vincenzo Librandi_, Feb 19 2015

%Y Cf. A116108, A116145, A126200, A126203, A163392, A163393, A253679, A253680, A253707, A253708, A002593, A253724.

%K nonn,easy

%O 2,1

%A _Vladimir Pletser_, Jan 10 2015