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A055418
Number of points in N^n of norm <= 3.
2
1, 4, 11, 29, 70, 157, 337, 702, 1420, 2780, 5258, 9615, 17043, 29381, 49430, 81404, 131563, 209084, 327237, 504945, 768820, 1155781, 1716375, 2518938, 3654750, 5244356, 7445244, 10461091, 14552809, 20051645, 27374612, 37042552, 49701157
OFFSET
0,2
LINKS
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
Row n=3 of A302998.
Cf. A055417 (case for norm <= 2).
Sequence in context: A153876 A036881 A275012 * A062432 A220018 A131046
KEYWORD
nonn,easy
STATUS
approved