

A089982


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


4



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

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


LINKS

Table of n, a(n) for n=1..48.


FORMULA

Triangular number m is in this sequence iff A000161(4*m+1)>1 or, alternatively, A083025(4*m+1)>1. [From 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 exist by considering a partial sum from x to y that is triangular, e.g. 15+16+17+18=66, then T(14)+T(11)=T(18). This particular example arises from 10+4k being triangular, e.g. 66 and we therefore have a solution.
All other solutions come from 3+2k, 6+3k, 10+4k, in general T(n)+nk being 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 *)


PROG

(PARI) { 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))); vecsort(y) }


CROSSREFS

Cf. A000217, A057100.
Sequence in context: A003340 A139606 A047717 * A151943 A207339 A056488
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



