OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
Satisfies a degree nine polynomial (see Example section). - Olivier Gérard, Mar 30 2015
G.f.: -(8*x^8-35*x^7+51*x^6-30*x^5-5*x^4+21*x^3-16*x^2+6*x-1) / (x-1)^10. - Colin Barker, Jul 07 2013
EXAMPLE
There are exactly 19 coordinate configurations (up to permutation) with up to 9 nonzero positive coordinates that can produce a vector of norm <= 3:
{..., 0, 0, 0, 0, 0, 0, 0, 0, 0} 0
{..., 0, 0, 0, 0, 0, 0, 0, 0, 1} 1
{..., 0, 0, 0, 0, 0, 0, 0, 0, 2} 2
{..., 0, 0, 0, 0, 0, 0, 0, 0, 3} 3
{..., 0, 0, 0, 0, 0, 0, 0, 1, 1} sqrt(2)
{..., 0, 0, 0, 0, 0, 0, 0, 1, 2} sqrt(5)
{..., 0, 0, 0, 0, 0, 0, 0, 2, 2} 2 sqrt(2)
{..., 0, 0, 0, 0, 0, 0, 1, 1, 1} sqrt(3)
{..., 0, 0, 0, 0, 0, 0, 1, 1, 2} sqrt(2) sqrt(3)
{..., 0, 0, 0, 0, 0, 0, 1, 2, 2} 3
{..., 0, 0, 0, 0, 0, 1, 1, 1, 1} 2
{..., 0, 0, 0, 0, 0, 1, 1, 1, 2} sqrt(7)
{..., 0, 0, 0, 0, 1, 1, 1, 1, 1} sqrt(5)
{..., 0, 0, 0, 0, 1, 1, 1, 1, 2} 2 sqrt(2)
{..., 0, 0, 0, 1, 1, 1, 1, 1, 1} sqrt(6)
{..., 0, 0, 0, 1, 1, 1, 1, 1, 2} 3
{..., 0, 0, 1, 1, 1, 1, 1, 1, 1} sqrt(7)
{..., 0, 1, 1, 1, 1, 1, 1, 1, 1} 2 sqrt(2)
{..., 1, 1, 1, 1, 1, 1, 1, 1, 1} 3
To produce the formula for a(n), it is sufficient to sum the number of permutations of these configurations in a vector of arbitrary length n.
This gives in the same order:
a(n) = 1 + n + n + n + binomial(n, 2) + n*(n - 1) + binomial(n, 2) + binomial(n, 3) + n*binomial(n-1, 2) + n*binomial(n-1, 2) + binomial(n, 4) + n*binomial(n-1, 3) + binomial(n, 5) + n*binomial(n-1, 4) + binomial(n, 6) + n*binomial(n-1, 5) + binomial(n, 7) + binomial(n, 8) + binomial(n, 9).
This is a polynomial of degree 9 in n.
a(n) = (1 + n) (9! + n (452016 + n (-224244 + n (152108 + n (-17351 + n (-16 + n (394 + (-28 + n) n)))))))/(9!).
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved