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

1,3

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.

Links

Table of n, a(n) for n=1..63.

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.

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, floor(-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