A302290 a(n) is the 2-norm of denominators of two-variable polynomials of degree n which are integer-valued.

1, 2, 1, 2, 4, 2, 1, 4, 5, 2, 4, 8, 4, 2, 5, 6, 5, 4, 8, 10, 5, 4, 9, 10, 4, 8, 16, 8, 4, 10, 9, 6, 9, 12, 12, 12, 9, 8, 13, 12, 8, 16, 20, 10, 9, 14, 13, 12, 12, 18, 21, 12, 9, 18, 20, 8, 16, 32, 16, 8, 20, 18, 9, 14, 25, 20, 16, 20, 17, 16, 17, 20, 24, 24, 24, 20, 17, 18, 21, 22, 20, 28, 29, 16, 17, 28, 24

Offset: 0

Author

Marie-Andree B.Langlois, Apr 04 2018

Keywords

• nonn

Comments

This is the 2-sequence of integer-valued polynomials of 2-variables. It can be shown that this also the 2-sequence of the homogeneous 3-variable integer valued polynomials where one of the variables is restricted to evaluate at odd values.

a(n) is also the n-th Bhargava's factorial when generalized to the two-variable case.

Links

• M. Bhargava, On P-orderings, Integer-Valued Polynomials, and Ultrametric Analysis, J. Amer. Math. Soc., 22 (2009), 963-993.
• S. Evrard, Bhargava's factorial in several variables, Journal of Algebra, 372 (2012), 134-148.

Crossrefs

Cf. A212429.

Formula

a(n) = 2^{k-1} if n = 2^k-k-1

a(2(2^k-k-1)-n) if 2^k-k-1 < n < 2^k-1

a(2(2^k-k-1)-n)+ 2a(n-2^k+1) if 2^k-1 <= n <= 2(2^k-k-1)

2a(n-2^k+1) if 2(2^k-k-1) < n < 2^{k+1}-k-2

where k is such that 2^k-k-1<= n.