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%I #50 Jan 18 2024 09:00:27
%S 2,11,45,65,109,280,644,1079,1309,3180,3355,6864,8284,9700,10681,
%T 10856,16775,17094,33506,35650,50435,63196,84330,105731,125220,145200,
%U 167986,220545,503690,678730,1046629,1065426
%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 i values.
%C a(n)^3 is the sum of two positive tetrahedrals A000292(j) + A000292(k). j values are A054209 and k values are A054210.
%C Another term (not necessarily the next one) is 2630045. - _R. J. Mathar_, Apr 07 2023
%H R. J. Mathar, <a href="/A054208/a054208.cxx.txt">C++ program for A054208</a>
%H Maciej Ulas, <a href="https://arxiv.org/abs/0811.2477">On certain diophantine equations related to triangular and tetrahedral numbers</a>, arXiv:0811.2477 [math.NT], 2008; Theorem 5.4.
%e 2^3 = 8 = binomial(2+2,3) + binomial(2+2,3).
%e 11^3 = 1331 = binomial(19+2,3) + binomial(3,3).
%p # is x(x+1)(x+2)/6= A000292(x)=n solvable?
%p # return true if yes.
%p isA000292 := proc(n)
%p local x;
%p if n = 0 then
%p return true ;
%p end if;
%p x := iroot(6*n,3) ;
%p # newton algorithm
%p while true do
%p x := x-round((x*(x+1)*(x+2)-6*n)/(3*x^2+6*x+2)) ;
%p if A000292(x) < n and A000292(x+1) >n then
%p return false ;
%p elif A000292(x) > n and A000292(x-1) <n then
%p return false ;
%p elif A000292(x) = n then
%p return true;
%p end if;
%p end do:
%p end proc:
%p isA054208 := proc(n)
%p local c,i,ti,tj;
%p c := n^3 ;
%p for i from 1 do
%p ti := A000292(i) ;
%p if ti > c/2 then
%p return false ;
%p end if ;
%p tj := c-ti ;
%p if isA000292(tj) then
%p return true ;
%p end if;
%p end do:
%p end proc:
%p for n from 1 do
%p if isA054208(n) then
%p print(n)
%p end if;
%p end do: # _R. J. Mathar_, Mar 17 2023
%t (* Range of j values is merely empirical *) jmin[k_] := Floor[Max[k, 1.86*k - 20000]]; jmax[k_] := Ceiling[1.86*k + 16000]; jmax[3005] = 10^5; ii = Reap[ Do[ Do[i = (Binomial[j+2, 3] + Binomial[k+2, 3])^(1/3); If[IntegerQ[i], Print[{i, j, k}]; Sow[i]; Break[]], {j, jmin[k], jmax[k]}], {k, 1, 40000}] ][[2, 1]]; A054208 = Union[ii] (* _Jean-François Alcover_, Dec 12 2012 *)
%Y Cf. A054209, A054210.
%K nice,nonn,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 Corrected by _T. D. Noe_, Oct 25 2006
%E a(21)-a(26) from _Sean A. Irvine_, Jan 25 2022
%E Terms complete up to 1065426 from _R. J. Mathar_, Apr 07 2023
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