|
%I #23 Oct 18 2021 16:55:47
%S 1,4,11,29,70,157,337,702,1420,2780,5258,9615,17043,29381,49430,81404,
%T 131563,209084,327237,504945,768820,1155781,1716375,2518938,3654750,
%U 5244356,7445244,10461091,14552809,20051645,27374612,37042552,49701157
%N Number of points in N^n of norm <= 3.
%H Andrew Howroyd, <a href="/A055418/b055418.txt">Table of n, a(n) for n = 0..1000</a>
%F Satisfies a degree nine polynomial (see Example section). - _Olivier Gérard_, Mar 30 2015
%F 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
%e There are exactly 19 coordinate configurations (up to permutation) with up to 9 nonzero positive coordinates that can produce a vector of norm <= 3:
%e {..., 0, 0, 0, 0, 0, 0, 0, 0, 0} 0
%e {..., 0, 0, 0, 0, 0, 0, 0, 0, 1} 1
%e {..., 0, 0, 0, 0, 0, 0, 0, 0, 2} 2
%e {..., 0, 0, 0, 0, 0, 0, 0, 0, 3} 3
%e {..., 0, 0, 0, 0, 0, 0, 0, 1, 1} sqrt(2)
%e {..., 0, 0, 0, 0, 0, 0, 0, 1, 2} sqrt(5)
%e {..., 0, 0, 0, 0, 0, 0, 0, 2, 2} 2 sqrt(2)
%e {..., 0, 0, 0, 0, 0, 0, 1, 1, 1} sqrt(3)
%e {..., 0, 0, 0, 0, 0, 0, 1, 1, 2} sqrt(2) sqrt(3)
%e {..., 0, 0, 0, 0, 0, 0, 1, 2, 2} 3
%e {..., 0, 0, 0, 0, 0, 1, 1, 1, 1} 2
%e {..., 0, 0, 0, 0, 0, 1, 1, 1, 2} sqrt(7)
%e {..., 0, 0, 0, 0, 1, 1, 1, 1, 1} sqrt(5)
%e {..., 0, 0, 0, 0, 1, 1, 1, 1, 2} 2 sqrt(2)
%e {..., 0, 0, 0, 1, 1, 1, 1, 1, 1} sqrt(6)
%e {..., 0, 0, 0, 1, 1, 1, 1, 1, 2} 3
%e {..., 0, 0, 1, 1, 1, 1, 1, 1, 1} sqrt(7)
%e {..., 0, 1, 1, 1, 1, 1, 1, 1, 1} 2 sqrt(2)
%e {..., 1, 1, 1, 1, 1, 1, 1, 1, 1}} 3
%e 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.
%e This gives in the same order:
%e 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).
%e This is a polynomial of degree 9 in n.
%e a(n) = (1 + n) (9! + n (452016 + n (-224244 + n (152108 + n (-17351 + n (-16 + n (394 + (-28 + n) n)))))))/(9!).
%Y Row n=3 of A302998.
%Y Cf. A055417 (case for norm <= 2).
%K nonn
%O 0,2
%A _David W. Wilson_
|
