A055418 Number of points in N^n of norm <= 3.

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

Author

David W. Wilson

Keywords

• nonn
• easy
• changed

Examples

			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!).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Row n=3 of A302998.

Cf. A055417 (case for norm <= 2).

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