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

1,1

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.

Links

Table of n, a(n) for n=1..59.

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

Example

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.

Crossrefs

Cf. A000217, A001110.

Sequence in context: A362399 A080823 A117925 * A138394 A140752 A246780

Adjacent sequences: A135568 A135569 A135570 * A135572 A135573 A135574

Keyword

nonn

Author

John W. Layman, Feb 23 2008

Status

approved