A176541 Numbers n such that there exist n consecutive triangular numbers which sum to a square.

0, 1, 2, 3, 4, 11, 13, 22, 23, 25, 27, 32, 37, 39, 46, 47, 48, 49, 50, 52, 59, 66, 71, 73, 83, 94, 98, 100, 104, 107, 109, 111, 118, 121, 128, 143, 146, 147, 148, 157, 167, 176, 179, 181, 183, 191, 192, 193, 194, 200, 214, 219, 227, 239, 241, 242, 243, 244, 253, 263

Offset: 1

Author

Andrew Weimholt, Apr 20 2010

Keywords

• nonn

Comments

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.

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.

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

Examples

			0 is in the sequence because the sum of 0 consecutive triangular numbers is 0 (a square).
1 is in the sequence because there exist triangular numbers which are squares (cf. A001110).
2 is in the sequence because ANY 2 consecutive triangular numbers sum to a square.
3 is in the sequence because there are infinitely many solutions (cf. A165517).
4 is in the sequence because there infinitely many solutions (cf. A202391).
5 is NOT in the sequence because no 5 consecutive triangular numbers sum to a square.
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.
For n=11, there exist infinitely many solutions (cf. A116476), so 11 is in the sequence.
		

Crossrefs

Cf. A176542, A000217, A000292, A001110, A077415.

Extensions

More terms from Max Alekseyev, May 10 2010