A127329 -id:A127329 - cp's OEIS frontend

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

Antonio Roldán, Sep 14 2017

nonn

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

			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.

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

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 *)

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)

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

A365081 Numbers k with the property that the symmetric representation of sigma(k) has four parts and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the diagram.

Original entry on oeis.org

21, 27, 33, 39, 51, 57, 69, 87, 93, 111, 123, 129, 141, 159, 177, 183, 201, 213, 219, 237, 249, 267, 291, 303, 309, 321, 327, 339, 381

Offset: 1

Omar E. Pol, Aug 20 2023

Also the row numbers of the triangle A364639 where the rows are [0, 0, 1, 0, -1, 1] or where the rows start with [0, 0, 1, 0, -1, 1] and the remaining terms are zeros.
Observation: the first 29 terms coincide with the first 29 terms of A161345 that are >= 21.
Apparently a(n)=A127329(n) for n>2. - R. J. Mathar, Sep 05 2023

			The symmetric representation of sigma(21) in the first quadrant looks like this:
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There are four parts (or polygons) and its second part is an octagon of width 1 and one of the vertices of the octagon is also the central vertex of the first valley of the largest Dyck path of the structure so 21 is in the sequence.

Subsequence of A016945, A264102, A280107 and A364414.

Cf. A033676, A161345, A196020, A235791, A236104, A237270 (parts), A237271, A237591, A237593, A240062, A245092, A249351 (widths), A262626, A364639.

cp's OEIS Frontend

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

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