A135571 Positive integers that are the difference of two positive triangular numbers in an odd number of ways.

2, 3, 4, 6, 8, 9, 10, 15, 16, 18, 21, 25, 28, 32, 45, 49, 50, 55, 64, 66, 72, 78, 81, 91, 98, 100, 105, 120, 121, 128, 136, 144, 153, 162, 169, 171, 190, 196, 200, 210, 225, 231, 242, 253, 256, 276, 288, 289, 300, 324, 325, 338, 351, 361, 378, 392, 400, 406, 435

Offset: 1

Author

John W. Layman, Feb 23 2008

Keywords

• nonn

Comments

Conjecture. This sequence is just the sequence of positive integers that are either square, twice a square, or triangular, but not both square and triangular (A001110). (This has been verified for n up to 100000.)

If the triangular number 0 is allowed, then Verhoeff has shown (see the reference) that the numbers that are the difference of two triangular numbers in exactly one way are just the powers of 2.

Examples

			As differences of two positive triangular numbers, 6 =21-15 (1 way), 9 =10-1 =15-6 =45-36 (3 ways), so 6 and 9 are terms of the sequence; 5 =6-1 = 15-10 (2 ways), so 5 is not a term of the sequence.
		

Links

• T. Verhoeff, Rectangular and Trapezoidal Arrangements, Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.6

Crossrefs

Cf. A000217, A001110.