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A089982 Triangular numbers that can be expressed as the sum of 2 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 analog 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. - 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) { 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, A012132, 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

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Last modified May 5 06:42 EDT 2016. Contains 272431 sequences.