%I #12 Jan 29 2020 09:46:02
%S 4,11,75,108,120,427,1309,1583,1753,2490,2712,2764,2822,3678,4502,
%T 4569,4595,7526,9667,13723,14279,18869,36367,37964,42669,43738,46820,
%U 52849,57666,59922,71592,80480,85480,96862,108383,121828,122426,131318,131398,155760,167021
%N Consider all integer triples (i,j,k), j,k>0, with i^3=j^3+binomial(k+2,3), ordered by increasing i; sequence gives i values.
%C j values are A054235 and k values are A054236
%H Jon E. Schoenfield, <a href="/A054234/b054234.txt">Table of n, a(n) for n=1..41</a>
%e 4^3=64=2^3+binomial(6+2,3); 11^3=1331=1^3+binomial(19+2,3)
%t nmax = 21;
%t ijk[0] = {1, 1, 0};
%t ijk[n_] := ijk[n] = Module[{i, j, k, r}, For[i = ijk[n-1][[1]]+1, True, i++, r = Reduce[0<j<i && k>0 && i^3 == j^3 + Binomial[k+2, 3], {j, k}, Integers]; If[r =!= False, Return[{i, j, k} /. ToRules[r]]]]];
%t triples = Reap[For[n=1, n<=nmax, n++, Print[ijk[n]]; Sow[ijk[n]]]][[2, 1]];
%t triples[[All, 1]] (* _Jean-François Alcover_, Jan 29 2020 *)
%K nice,nonn
%O 1,1
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Feb 07 2000
%E More terms from _Jon E. Schoenfield_, Jan 19 2009