A292309 Numbers equal to the sum of three triangular numbers in arithmetic progression.

9, 63, 84, 108, 234, 315, 459, 513, 570, 630, 759, 975, 1053, 1134, 1395, 1584, 1998, 2109, 2709, 2838, 2970, 3105, 3384, 3528, 3825, 4134, 4455, 4620, 4788, 4959, 5133, 5310, 5673, 5859, 6834, 7038, 7668, 7884, 8325, 8778, 9009, 9243, 9480, 10209, 10710, 11223

Offset: 1

Author

Antonio Roldán, Sep 14 2017

Keywords

• nonn

Comments

Subsequence of A045943, because a(n) = 3*k*(k+1)/2 = 3*A000217(k) for some k.

Examples

			9 = A000217(0) + A000217(2) + A000217(3) = 0 + 3 + 6, with 6 - 3 = 3 - 0 = 3.
513 = A000217(11) + A000217(18) + A000217(23) = 66 + 171 + 276, with 171 - 66 = 276 - 171 = 105.
		

Crossrefs

Cf. A000217, A045943, A127329, A292310, A292313, A292314, A292316.

Programs

Mathematica

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 *)

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)

Formula

a(n) = 3*A292310(n).