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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.
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%I #21 May 07 2024 12:08:07

%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 262224,286321,335103,904043,1190328,1901197,1833590

%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 *)

%t A054208 = Cases[Import["https://oeis.org/A054208/b054208.txt", "Table"], {_, _}][[All, 2]];

%t 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}];

%t A054209 = ijk[[All, 2]] (* _Jean-François Alcover_, May 07 2024 *)

%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

%E a(27)-a(31) from _Jean-François Alcover_, May 07 2024