

A210449


Numbers that are the sum of three triangular numbers an odd number of ways.


0



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
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OFFSET

1,3


COMMENTS

Reduce the elements of A192717 by subtracting 3 and dividing by 8. This makes sense since the elements of A192717 are congruent to 3 (mod 8).
A positive integer n belongs to this sequence precisely when n can be written as t + 2u for triangular numbers t, u an odd number of times, equivalently, written as t + u + v for triangular numbers t, u, v, an odd number of times.


REFERENCES

J. Cooper, D. Eichhorn, and K. O'Bryant, Reciprocals of binary power series, International Journal of Number Theory, 2 no. 4 (2006), 499522.


LINKS

Table of n, a(n) for n=1..63.
J. N. Cooper and A. W. N. Riasanovsky, On the Reciprocal of the Binary Generating Function for the Sum of Divisors, 2012


EXAMPLE

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.
...


PROG

(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, (1+sqrt(8*prec+1))/2)]
tSeries = BPS(tList)
print (tSeries^3).exponents()[:128]


CROSSREFS

Cf. A192717, A192628.
Sequence in context: A195997 A186277 A061770 * A080639 A186306 A047483
Adjacent sequences: A210446 A210447 A210448 * A210450 A210451 A210452


KEYWORD

nonn


AUTHOR

Alexander Riasanovsky, Jan 20 2013


STATUS

approved



