A210449 Numbers that are the sum of three triangular numbers an odd number of ways.
0, 1, 2, 5, 7, 8, 9, 10, 12, 13, 16, 17, 18, 20, 21, 22, 26, 28, 30, 31, 34, 35, 38, 41, 43, 45, 47, 48, 52, 55, 58, 59, 61, 62, 63, 65, 66, 67, 68, 70, 71, 73, 75, 77, 80, 82, 85, 86, 92, 93, 98, 101, 103, 107, 108, 110, 111, 113, 116, 118, 120, 121, 127
Offset: 1
Keywords
Examples
For n = 0, 1 representation: 0 + 0 + 0; so 0 belongs to this sequence. For n = 1, 3 representations: 1 + 0 + 0, 0 + 1 + 0, 0 + 0 + 1; so 1 belongs. For n = 2, 3 representations: 1 + 1 + 0, 1 + 0 + 1, 0 + 1 + 1; so 2 belongs. For n = 3, 4 representations: 3 + 0 + 0, 0 + 3 + 0, 0 + 0 + 3, 1 + 1 + 1; so 3 does not belong. For n = 4, 6 representations: 3 + 1 + 0, 3 + 0 + 1, 1 + 3 + 0, 1 + 0 + 3, 0 + 3 + 1, 0 + 1 + 3; so 4 does not belong. ...
Links
- J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, arXiv:math/0506496 [math.NT], 2005.
- J. N. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499-522.
- J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012.
Programs
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Sage
def BPS(n): #binary power series return sum([q^s for s in n]) prec = 2^14 R = PowerSeriesRing(GF(2), 'q', default_prec = prec) q = R.gen() tList = [(n*(n+1))//2 for n in range(0, floor(-1+sqrt(8*prec+1))//2)] tSeries = BPS(tList) print((tSeries^3).exponents()[:128])
Comments