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A135571 Positive integers that are the difference of two positive triangular numbers in an odd number of ways. 1

%I #2 Mar 30 2012 17:37:54

%S 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,

%T 100,105,120,121,128,136,144,153,162,169,171,190,196,200,210,225,231,

%U 242,253,256,276,288,289,300,324,325,338,351,361,378,392,400,406,435

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

%C 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.)

%C 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.

%H T. Verhoeff, <a href="http://www.cs.uwaterloo.ca/journals/JIS/trapzoid.html">Rectangular and Trapezoidal Arrangements, Journal of Integer Sequences, Vol. 2 (1999), Article 99.1.6</a>

%e 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.

%Y Cf. A000217, A001110.

%K nonn

%O 1,1

%A _John W. Layman_, Feb 23 2008

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