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A135571
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Positive integers that are the difference of two positive triangular numbers in an odd number of ways.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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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
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EXAMPLE
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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.
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CROSSREFS
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Cf. A000217, A001110.
Sequence in context: A303434 A080823 A117925 * A138394 A140752 A246780
Adjacent sequences: A135568 A135569 A135570 * A135572 A135573 A135574
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KEYWORD
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nonn
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AUTHOR
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John W. Layman, Feb 23 2008
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STATUS
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approved
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