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%I #23 Jul 11 2016 20:05:07
%S 2,3,5,6,7,8,10,12,13,18,20,27,28,33,37,42,45,47,55,58,60,62,63,65,67,
%T 73,75,78,80,85,88,90,92,102,103,105,112,115,118,120,125,128,130,132,
%U 135,140,142,150,153,157,163,170,175,192,193,198,200,203,210,215,218,220,222
%N Numbers whose square is the sum of two positive triangular numbers in exactly one way.
%C Obviously, A000217(n) + A000217(n+1) = n*(n+1)/2 + (n+1)*(n+2)/2 = (n+1)^2. So every square that is greater than 1 is the sum of two positive consecutive triangular numbers. This sequence focuses on the squares that have only this trivial solution.
%C For a related comment, see comments section of A001912.
%e 3 is a term because 3^2 is the sum of two positive triangular numbers in exactly 1 way that is: 3^2 = 3 + 6.
%t nR[n_]:= (SquaresR[2, n]+Plus@@ Pick[{-4, 4}, IntegerQ/@ Sqrt[{n, n/2}]])/8 ; nTr[n_] := nR[8*n + 2] - Boole@ IntegerQ@ Sqrt[8*n + 1]; Select[Range[250], nTr[#^2]==1 &] (* _Giovanni Resta_, Jul 08 2016 *)
%Y Cf. A000217, A000290, A001912, A230312, A274758.
%K nonn,easy
%O 1,1
%A _Altug Alkan_, Jul 06 2016
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