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
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
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