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

1,1

Comments

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

Links

Table of n, a(n) for n=1..46.

Formula

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

Example

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.

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 *)

Prog

(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)

Crossrefs

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

Sequence in context: A341627 A159235 A181403 * A337236 A285456 A085645

Adjacent sequences: A292306 A292307 A292308 * A292310 A292311 A292312

Keyword

nonn

Author

Antonio Roldán, Sep 14 2017

Status

approved