A112352 Triangular numbers that are the sum of two distinct positive triangular numbers.

21, 36, 55, 66, 91, 120, 136, 171, 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, 2926

Offset: 1

Author

Rick L. Shepherd, Sep 05 2005

Keywords

• nonn

Comments

Subsequence of A089982: it doesn't require the two positive triangular numbers to be distinct.

Subsequence of squares: 36, 1225, 41616, 1413721,... is also in A001110. - Zak Seidov, May 07 2015

First term with 2 representations is 231: 21+210=78+153, first term with 3 representations is 276: 45+211=66+120=105+171; apparently the number of representations is unbounded. - Zak Seidov, May 11 2015

Examples

			36 is a term because 36 = 15 + 21 and these three numbers are distinct triangular numbers (A000217(8) = A000217(5) + A000217(6)).
		

Links

• Zak Seidov, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000217 (triangular numbers), A112353 (triangular numbers that are the sum of three distinct positive triangular numbers), A089982.

Cf. A001110. - Zak Seidov, May 07 2015

Programs

Maple

N:= 10^5: # to get all terms <= N
S:= {}:
for a from 1 to floor(sqrt(1+8*N)/2) do
  for b from 1 to a-1 do
    y:= a*(a+1)/2 + b*(b+1)/2;
      if y > N then break fi;
      if issqr(8*y+1) then S:= S union {y} fi
  od
od:
sort(convert(S,list)); # Robert Israel, May 13 2015

Mathematica

Select[Union[Total/@Subsets[Accumulate[Range[100]],{2}]],OddQ[ Sqrt[ 1+8#]]&] (* Harvey P. Dale, Feb 28 2016 *)

Extensions

Offset corrected by Arkadiusz Wesolowski, Aug 06 2012