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Triangular numbers that are the sum of two distinct positive triangular numbers.
3

%I #23 Feb 28 2016 11:23:31

%S 21,36,55,66,91,120,136,171,231,276,351,378,406,496,561,666,703,741,

%T 820,861,946,990,1035,1081,1176,1225,1326,1378,1431,1485,1540,1596,

%U 1653,1711,1770,1891,1953,2016,2080,2211,2278,2346,2556,2701,2775,2850,2926

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

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

%C Subsequence of squares: 36, 1225, 41616, 1413721,... is also in A001110. - _Zak Seidov_, May 07 2015

%C 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 Zak Seidov, <a href="/A112352/b112352.txt">Table of n, a(n) for n = 1..1000</a>

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

%p N:= 10^5: # to get all terms <= N

%p S:= {}:

%p for a from 1 to floor(sqrt(1+8*N)/2) do

%p for b from 1 to a-1 do

%p y:= a*(a+1)/2 + b*(b+1)/2;

%p if y > N then break fi;

%p if issqr(8*y+1) then S:= S union {y} fi

%p od

%p od:

%p sort(convert(S,list)); # _Robert Israel_, May 13 2015

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

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

%Y Cf. A001110. - _Zak Seidov_, May 07 2015

%K nonn

%O 1,1

%A _Rick L. Shepherd_, Sep 05 2005

%E Offset corrected by _Arkadiusz Wesolowski_, Aug 06 2012