%I #16 Jan 25 2022 08:46:38
%S 2,19,74,113,197,482,1162,1959,1937,5644,6061,10788,12772,17624,19401,
%T 16503,29195,25487,60881,63348,89133,114519,140524,192059,214754,
%U 262224,286321
%N Consider all integer triples (i,j,k), j >= k>0, with i^3=binomial(j+2,3)+binomial(k+2,3), ordered by increasing i; sequence gives j values.
%C i values are A054208 and k values are A054210.
%e 2^3 = 8 = binomial(2+2,3) + binomial(2+2,3).
%e 11^3 = 1331 = binomial(19+2,3) + binomial(3,3).
%t (* This is just a re-computation from A054208 *) A054208 = {2, 11, 45, 65, 109, 280, 644, 1079, 1309, 3180, 3355, 6864, 8284, 9700, 10681, 10856, 16775, 17094, 33506, 35650, 50435}; ijk = Table[ sol = {i, j, k} /. ToRules[ Reduce[ 0 < k <= j && 6*i^3 == j*(j+1)*(j+2) + k*(k+1)*(k+2), {j, k}, Integers]]; Print[sol]; sol, {i, A054208 }]; A054209 = ijk[[All, 2]] (* _Jean-François Alcover_, Sep 11 2012 *)
%Y Cf. A054208, A054210.
%K nonn,nice,more
%O 0,1
%A Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Jan 31 2000
%E More terms from _Sascha Kurz_, Mar 22 2002
%E a(21)-a(26) from _Sean A. Irvine_, Jan 25 2022
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