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%I #56 Aug 20 2022 02:24:45
%S 1,15,8,84,27,249,64,552,125,1035,216,1740,343,2709,512,3984,729,5607,
%T 1000,7620,1331,10065,1728,12984,2197,16419,2744,20412,3375,25005,
%U 4096,30240,4913,36159,5832,42804,6859,50217,8000,58440,9261,67515,10648,77484,12167,88389,13824
%N Sequence of distinct least positive numbers such that the average of the first n terms is a cube.
%C Colin Barker's formulas are true if the curve x^3 = 7*y^3 + 6*y^2 + 2*y has no positive integer solutions. This is a curve of genus 1 (equivalent to the elliptic curve s^3 + t^2 + 20), and does have some rational points, but no positive integer solutions at least for y <= 10^21. - _Robert Israel_, May 17 2015
%C Now confirmed: that curve has no positive integer solutions. See the Mathematics Stack Exchange link. - _Robert Israel_, May 18 2015
%H Colin Barker, <a href="/A245624/b245624.txt">Table of n, a(n) for n = 1..1000</a>
%H R. Israel, W. Jagy and Á. Lozano-Robledo, <a href="http://math.stackexchange.com/questions/1283013/integer-solutions-of-x3-7y3-6-y22-y">Integer Solutions of x^3 = 7 y^3 + 6 y^2 + 2 y</a>, Mathematics Stack Exchange question (2015).
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,4,0,-6,0,4,0,-1).
%F a(2*n-1) = n^3, a(2*n) = 7*n^3 + 6*n^2 + 2*n.
%F a(n) = 4*a(n-2) - 6*a(n-4) + 4*a(n-6) - a(n-8) for n > 8. - _Colin Barker_, Nov 05 2014
%F G.f.: x*(3*x^5 + x^4 + 24*x^3 + 4*x^2 + 15*x + 1) / ((x-1)^4*(x+1)^4). - _Colin Barker_, Nov 05 2014
%p seq(op([k^3, 7*k^3+6*k^2+2*k]),k=1..100); # _Robert Israel_, May 18 2015
%t Flatten[Table[{n^3, 7 n^3 + 6 n^2 + 2 n}, {n, 25}]] (* _Vincenzo Librandi_, May 19 2015 *)
%o (PARI) v=[]; n=1; while(n<10^5, num=(vecsum(v)+n); if(num%(#v+1)==0&&vecsearch(vecsort(v), n)==0, for(i=1, n+2, if(i^3>(num/(#v+1)), break); if(i^3==(num/(#v+1)), print1(n, ", "); v=concat(v, n); n=1; break))); n++)
%o (Magma) &cat[[k^3, 7*k^3+6*k^2+2*k]: k in [1..25]]; // _Vincenzo Librandi_, May 19 2015
%o (PARI) Vec(x*(3*x^5+x^4+24*x^3+4*x^2+15*x+1)/((x-1)^4*(x+1)^4) + O(x^100)) \\ _Colin Barker_, May 19 2015
%Y Cf. A085047, A245621.
%K nonn,easy
%O 1,2
%A _Derek Orr_, Nov 05 2014
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