cp's OEIS Frontend

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

%I A055418 #29 Aug 31 2025 09:11:19
%S A055418 1,4,11,29,70,157,337,702,1420,2780,5258,9615,17043,29381,49430,81404,
%T A055418 131563,209084,327237,504945,768820,1155781,1716375,2518938,3654750,
%U A055418 5244356,7445244,10461091,14552809,20051645,27374612,37042552,49701157
%N A055418 Number of points in N^n of norm <= 3.
%H A055418 Andrew Howroyd, <a href="/A055418/b055418.txt">Table of n, a(n) for n = 0..1000</a>
%H A055418 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A055418 Satisfies a degree nine polynomial (see Example section). - _Olivier Gérard_, Mar 30 2015
%F A055418 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 A055418 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 A055418 {..., 0, 0, 0, 0, 0, 0, 0, 0, 0}   0
%e A055418 {..., 0, 0, 0, 0, 0, 0, 0, 0, 1}   1
%e A055418 {..., 0, 0, 0, 0, 0, 0, 0, 0, 2}   2
%e A055418 {..., 0, 0, 0, 0, 0, 0, 0, 0, 3}   3
%e A055418 {..., 0, 0, 0, 0, 0, 0, 0, 1, 1}   sqrt(2)
%e A055418 {..., 0, 0, 0, 0, 0, 0, 0, 1, 2}   sqrt(5)
%e A055418 {..., 0, 0, 0, 0, 0, 0, 0, 2, 2}   2 sqrt(2)
%e A055418 {..., 0, 0, 0, 0, 0, 0, 1, 1, 1}   sqrt(3)
%e A055418 {..., 0, 0, 0, 0, 0, 0, 1, 1, 2}   sqrt(2) sqrt(3)
%e A055418 {..., 0, 0, 0, 0, 0, 0, 1, 2, 2}   3
%e A055418 {..., 0, 0, 0, 0, 0, 1, 1, 1, 1}   2
%e A055418 {..., 0, 0, 0, 0, 0, 1, 1, 1, 2}   sqrt(7)
%e A055418 {..., 0, 0, 0, 0, 1, 1, 1, 1, 1}   sqrt(5)
%e A055418 {..., 0, 0, 0, 0, 1, 1, 1, 1, 2}   2 sqrt(2)
%e A055418 {..., 0, 0, 0, 1, 1, 1, 1, 1, 1}   sqrt(6)
%e A055418 {..., 0, 0, 0, 1, 1, 1, 1, 1, 2}   3
%e A055418 {..., 0, 0, 1, 1, 1, 1, 1, 1, 1}   sqrt(7)
%e A055418 {..., 0, 1, 1, 1, 1, 1, 1, 1, 1}   2 sqrt(2)
%e A055418 {..., 1, 1, 1, 1, 1, 1, 1, 1, 1}   3
%e A055418 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 A055418 This gives in the same order:
%e A055418 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 A055418 This is a polynomial of degree 9 in n.
%e A055418 a(n) = (1 + n) (9! + n (452016 + n (-224244 + n (152108 + n (-17351 + n (-16 + n (394 + (-28 + n) n)))))))/(9!).
%Y A055418 Row n=3 of A302998.
%Y A055418 Cf. A055417 (case for norm <= 2).
%K A055418 nonn,easy,changed
%O A055418 0,2
%A A055418 _David W. Wilson_