A055427 Number of points in Z^n of norm <= 3.

1, 7, 29, 123, 425, 1343, 4197, 12435, 33809, 84663, 198765, 444907, 959801, 2005615, 4064821, 7988867, 15221537, 28122727, 50423741, 87851099, 148962249, 246243487, 397527813, 627798387, 971451697, 1475103511, 2201030157

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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 A302997. Column 9 of A122510.

Programs

Mathematica

a[n_] := SeriesCoefficient[1/(1-x) Sum[x^(i^2), {i, -3, 3}]^n, {x, 0, 9}];
a /@ Range[0, 26] (* Jean-François Alcover, Sep 29 2019, from A302997 *)

Formula

Appears to satisfy a 9-degree polynomial. - Ralf Stephan, Mar 07 2004

Empirical g.f.: (x+1)*(93*x^8+620*x^7-848*x^6+516*x^5-150*x^4+20*x^3+8*x^2-4*x+1) / (x-1)^10. - Colin Barker, Jul 07 2013

Above conjectures confirmed by later additions of b-file from Andrew Howroyd and program from Jean-François Alcover with connection to A302997. - Ray Chandler, Jun 27 2024

a(n) = 1 +1126/315*n +84668/2835*n^3 -418/45*n^2 +2152/135*n^5 -64/15*n^6 +584/945*n^7 -2/45*n^8 -152/5*n^4 +4/2835*n^9. - R. J. Mathar, Aug 03 2025