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%I #17 Jan 25 2022 08:48:03
%S 2,1,49,54,19,266,308,197,1834,2354,1562,8812,10988,998,1959,14706,
%T 15089,23758,3005,26023,39490,23156,93724,19401,123338,69550,170653
%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 k values.
%C i values are A054208 and j values are A054209.
%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 }]; A054210 = ijk[[All, 3]] (* _Jean-François Alcover_, Sep 11 2012 *)
%Y Cf. A054208, A054209.
%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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