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%I #24 Apr 01 2018 13:10:23
%S 0,1,2,3,4,11,13,22,23,25,27,32,37,39,46,47,48,49,50,52,59,66,71,73,
%T 83,94,98,100,104,107,109,111,118,121,128,143,146,147,148,157,167,176,
%U 179,181,183,191,192,193,194,200,214,219,227,239,241,242,243,244,253,263
%N Numbers n such that there exist n consecutive triangular numbers which sum to a square.
%C Numbers n such that there exists some x >= 0 such that A000292(x+n) - A000292(x) is a square. Terms of this sequence, for which only a finite number of solutions x exist, are given in A176542.
%C Integer n is in the sequence if there exist non-degenerate solutions to the Diophantine equation: 8x^2 - n*y^2 - A077415(n) = 0. A degenerate solution is one involving triangular numbers with negative indexes.
%C The sum of n consecutive triangular numbers starting at the j-th is Sum_{k=j..j+n-1} A000217(k) = n*(n^2 + 3*j*n + 3*j^2 - 1)/6, see A143037. - _R. J. Mathar_, May 06 2015
%e 0 is in the sequence because the sum of 0 consecutive triangular numbers is 0 (a square).
%e 1 is in the sequence because there exist triangular numbers which are squares (cf. A001110).
%e 2 is in the sequence because ANY 2 consecutive triangular numbers sum to a square.
%e 3 is in the sequence because there are infinitely many solutions (cf. A165517).
%e 4 is in the sequence because there infinitely many solutions (cf. A202391).
%e 5 is NOT in the sequence because no 5 consecutive triangular numbers sum to a square.
%e For n=8, solutions to the Diophantine equation exist, but start at A000217(-2) and A000217(-6): 1 + 0 + 0 + 1 + 3 + 6 + 10 + 15 = 36 and 15 + 10 + 6 + 3 + 1 + 0 + 0 + 1 = 36. There are no non-degenerate solutions for n=8. Hence, 8 is not included in the sequence.
%e For n=11, there exist infinitely many solutions (cf. A116476), so 11 is in the sequence.
%Y Cf. A176542, A000217, A000292, A001110, A077415.
%K nonn
%O 1,3
%A _Andrew Weimholt_, Apr 20 2010
%E More terms from _Max Alekseyev_, May 10 2010
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