cp's OEIS Frontend

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

%I A112352 #23 Feb 28 2016 11:23:31
%S A112352 21,36,55,66,91,120,136,171,231,276,351,378,406,496,561,666,703,741,
%T A112352 820,861,946,990,1035,1081,1176,1225,1326,1378,1431,1485,1540,1596,
%U A112352 1653,1711,1770,1891,1953,2016,2080,2211,2278,2346,2556,2701,2775,2850,2926
%N A112352 Triangular numbers that are the sum of two distinct positive triangular numbers.
%C A112352 Subsequence of A089982: it doesn't require the two positive triangular numbers to be distinct.
%C A112352 Subsequence of squares: 36, 1225, 41616, 1413721,... is also in A001110. - _Zak Seidov_, May 07 2015
%C A112352 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
%H A112352 Zak Seidov, <a href="/A112352/b112352.txt">Table of n, a(n) for n = 1..1000</a>
%e A112352 36 is a term because 36 = 15 + 21 and these three numbers are distinct triangular numbers (A000217(8) = A000217(5) + A000217(6)).
%p A112352 N:= 10^5: # to get all terms <= N
%p A112352 S:= {}:
%p A112352 for a from 1 to floor(sqrt(1+8*N)/2) do
%p A112352   for b from 1 to a-1 do
%p A112352     y:= a*(a+1)/2 + b*(b+1)/2;
%p A112352       if y > N then break fi;
%p A112352       if issqr(8*y+1) then S:= S union {y} fi
%p A112352   od
%p A112352 od:
%p A112352 sort(convert(S,list)); # _Robert Israel_, May 13 2015
%t A112352 Select[Union[Total/@Subsets[Accumulate[Range[100]],{2}]],OddQ[ Sqrt[ 1+8#]]&] (* _Harvey P. Dale_, Feb 28 2016 *)
%Y A112352 Cf. A000217 (triangular numbers), A112353 (triangular numbers that are the sum of three distinct positive triangular numbers), A089982.
%Y A112352 Cf. A001110. - _Zak Seidov_, May 07 2015
%K A112352 nonn
%O A112352 1,1
%A A112352 _Rick L. Shepherd_, Sep 05 2005
%E A112352 Offset corrected by _Arkadiusz Wesolowski_, Aug 06 2012