A089982 Triangular numbers that can be expressed as the sum of 2 positive triangular numbers.

6, 21, 36, 55, 66, 91, 120, 136, 171, 210, 231, 276, 351, 378, 406, 496, 561, 666, 703, 741, 820, 861, 946, 990, 1035, 1081, 1176, 1225, 1326, 1378, 1431, 1485, 1540, 1596, 1653, 1711, 1770, 1891, 1953, 2016, 2080, 2211, 2278, 2346, 2556, 2701, 2775, 2850

Offset

1,1

Comments

Intersection of triangular numbers with sumset of triangular numbers. Triangular number analog of what for squares is {A057100(n)^2} = {A009000(n)^2}. {A000217} INTERSECT {A000217 + A000217}. - Jonathan Vos Post, Mar 09 2007

A subsequence of A051533. - Wolfdieter Lang, Jan 11 2017

Links

Michael S. Branicky, Table of n, a(n) for n = 1..11915

Formula

Triangular number m is in this sequence iff A000161(4*m+1)>1 or, alternatively, A083025(4*m+1)>1. - Max Alekseyev, Oct 24 2008

a(n) = A000217(A012132(n)). - Ivan N. Ianakiev, Jan 17 2013

Example

Generally, A000217(A000217(n)) = A000217(A000217(n)-1) + A000217(n) and so is automatically included. These are 6=T(3), 21=T(6), 55=T(10), etc. Other solutions occur when a partial sum from x to y is triangular, e.g., 15 + 16 + 17 + 18 = 66 = T(11), so T(14) + T(11) = T(18). This particular example arises since 10+4k is triangular (at k=14, 10 + 4k = 66), and we therefore have a solution.
All other solutions occur when 3+2k, 6+3k, 10+4k, etc. -- in general, T(j) + j*k -- is triangular.

Mathematica

trn[i_]:=Module[{trnos=Accumulate[Range[i]], t2s}, t2s=Union[Total/@ Tuples[ trnos, 2]]; Intersection[trnos, t2s]] (* Harvey P. Dale, Nov 08 2011 *)
Select[Range[75], ! PrimeQ[#^2 + (# + 1)^2] &] /. Integer_ -> (Integer^2 + Integer)/2 (* Arkadiusz Wesolowski, Dec 03 2015 *)

Prog

(PARI) t(i) = i*(i+1)/2;
{ v=vector(100, i, t(i)); y=vector(100); c=0; for (i=1, 30, for (j=i, 30, x=t(i)+t(j); f=0; for (k=1, 100, if (x==v[k], f=1; break)); if (f==1, y[c++ ]=x))); select(x->(x>0), vecsort(y, , 8)) } \\ slightly edited by Michel Marcus, Apr 15 2021
(PARI) lista(nn) = {for (n=1, nn, my(t = n*(n+1)/2); for (k=1, n-1, if (ispolygonal(t - k*(k+1)/2, 3), print1(t, ", "); break; )); ); } \\ Michel Marcus, Apr 15 2021
(Python)
from itertools import count, takewhile
def aupto(lim):
    t = list(takewhile(lambda x: x<=lim, (i*(i+1)//2 for i in count(1))))
    s = set(a+b for i, a in enumerate(t) for b in t[i:])
    return sorted(s & set(t))
print(aupto(3000)) # Michael S. Branicky, Jun 21 2021

Crossrefs

Cf. A000217, A012132, A051533, A057100.

Sequence in context: A345813 A139606 A047717 * A151943 A207339 A284988

Adjacent sequences: A089979 A089980 A089981 * A089983 A089984 A089985

Keyword

nonn,easy

Author

Jon Perry, Jan 13 2004

Extensions

More terms from Lambert Klasen (Lambert.Klasen(AT)gmx.net) and David Wasserman, Sep 23 2005

Status

approved