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%I #22 Jun 25 2023 05:46:56
%S 9,63,84,108,234,315,459,513,570,630,759,975,1053,1134,1395,1584,1998,
%T 2109,2709,2838,2970,3105,3384,3528,3825,4134,4455,4620,4788,4959,
%U 5133,5310,5673,5859,6834,7038,7668,7884,8325,8778,9009,9243,9480,10209,10710,11223
%N Numbers equal to the sum of three triangular numbers in arithmetic progression.
%C Subsequence of A045943, because a(n) = 3*k*(k+1)/2 = 3*A000217(k) for some k.
%F a(n) = 3*A292310(n).
%e 9 = A000217(0) + A000217(2) + A000217(3) = 0 + 3 + 6, with 6 - 3 = 3 - 0 = 3.
%e 513 = A000217(11) + A000217(18) + A000217(23) = 66 + 171 + 276, with 171 - 66 = 276 - 171 = 105.
%t Module[{t = 3, k = 2, i, e, v, m}, Reap[While[t <= 5000, i = k; e = 0; v = t+i; While[i > 0 && e == 0, If[IntegerQ @ Sqrt[8v+1], m = 3t; e = 1; Sow[m]]; i--; v += i]; k++; t += k]][[2, 1]]] (* _Jean-François Alcover_, Jun 25 2023, after PARI code *)
%o (PARI) t=3;k=2;while(t<=5000,i=k;e=0;v=t+i;while(i>1&&e==0,if(issquare(8*v+1),m=3*t;e=1;print1(m,", "));i+=-1;v+=i);k+=1;t+=k)
%Y Cf. A000217, A045943, A127329, A292310, A292313, A292314, A292316.
%K nonn
%O 1,1
%A _Antonio Roldán_, Sep 14 2017
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