A072521 a(1) = 6 and then the smallest triangular numbers, larger than the previous term, such that sum of two neighbors is also a triangular number.

6, 15, 21, 45, 91, 990, 1711, 365085, 401856, 713415, 785631, 1079715, 1326006, 2355535, 2888406, 5137615, 5666661, 5764710, 9550635, 9921285, 10934826, 19434495, 21421785, 23622501, 42003195, 46315500, 82349361, 146384605, 152259975, 171578550, 288300078, 313137825

Offset

1,1

Comments

The sequence is unbounded as a(n+1) is at most the a(n)-th triangular number.

If a(n) + u*(u+1)/2 = t*(t+1)/2 then 2*a(n) = (t-u)*(t+u+1) so t-u and t+u+1 are divisors of 2*a(n) and t-u and t+u+1 have different parity. - David A. Corneth, Jan 06 2026

Links

Robert Israel, Table of n, a(n) for n = 1..359

Example

45 is a term because 21 + 45 = 66, 45 + 91 = 136, and 66 and 136 are triangular numbers.

Maple

R:= 6: r:= 6: s:= 3:
for i from 1 to 30 do
  s:= min(select(`>`, map(subs, {isolve(1 + 8*r + 4*y*(y+1) = z^2)}, y), s));
  r:= s*(s+1)/2;
  R:= R, r
od:
R; # Robert Israel, Jan 05 2026

Prog

(PARI) p=6; k=3; print1(p", "); for(n=1, 30, k=k+1; u=p+k*(k+1)/2; t=sqrtint(2*u); while(u!=t*(t+1)/2, k=k+1; u=p+k*(k+1)/2; t=sqrtint(2*u)); p=k*(k+1)/2; print1(p", "))
(PARI) first(n) = {my(res=List([6])); for(i=2, n-1, listput(res, nxt(res[#res]))); res}
nxt(n) = {my(d = divisors(2*n), u); forstep(i = (#d + 1)\2, 1, -1, if(bitand(d[i]+d[#d + 1 - i], 1), u = (d[#d + 1 - i] - 1 - d[i])/2; c = u*(u+1)/2; if(c > n, return(c))))} \\ David A. Corneth, Jan 06 2026

Crossrefs

Cf. A072522.

Sequence in context: A332877 A357325 A362211 * A130178 A100410 A095032

Adjacent sequences: A072518 A072519 A072520 * A072522 A072523 A072524

Keyword

nonn

Author

Amarnath Murthy, Jul 31 2002

Extensions

More terms from Ralf Stephan, Mar 30 2003

Name clarified by David A. Corneth, Jan 05 2026

Status

approved