

A123999


Number of ordered ways of writing n as a sum of 4 squares of nonnegative numbers less than 4.


1



1, 4, 6, 4, 5, 12, 12, 4, 6, 16, 18, 12, 8, 16, 24, 12, 0, 12, 18, 12, 6, 4, 12, 12, 0, 0, 6, 4, 4, 0, 0, 4, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
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OFFSET

0,2


COMMENTS

Through n = 15, a(n) = number of ordered ways to write n as the sum of 4 squares. For n > 15, we must exclude sums which include 4^2, 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 16, 24, 25, 29, 30, 32, 33, 34, 35 and all n > 36. Without the restriction on the size of squares, all natural numbers can be written as the sum of 4 squares, as Lagrange proved in 1750. This sequence is to 4 as A123337 Number of ordered ways to write n as the sum of 5 squares less than 5, is to 5.


LINKS

Table of n, a(n) for n=0..72.


FORMULA

a(n) = Card{(a,b,c,d) such that 0<=a,b,c,d<4 and a^2 + b^2 + c^2 + d^2 = n}.


EXAMPLE

a(0) = 1 because of the unique sum 0 = 0^2 + 0^2 + 0^2 + 0^2.
a(1) = 4 because of the 4 permutations 1 = 0^2 + 0^2 + 0^2 + 1^2 = 0^2 + 0^2 + 1^2 + 0^2 = 0^2 + 1^2 + 0^2 + 0^2 = 1^2 + 0^2 + 0^2 + 0^2.
a(4) = 5 because of 4 = 1^2 + 1^2 + 1^2 + 1^2 plus the 4 permutations of 4 = 0^2 + 0^2 + 0^2 + 2^2.
a(16) = 0 because we must, by definition, exclude 16 = 2^2 + 2^2 + 2^2 + 2^2 and no other sum of exactly 4 squares totals 16.


CROSSREFS

Cf. A000118, A014110, A123337.
Sequence in context: A132024 A092039 A243371 * A014110 A091651 A199865
Adjacent sequences: A123996 A123997 A123998 * A124000 A124001 A124002


KEYWORD

easy,nonn


AUTHOR

Jonathan Vos Post, Oct 31 2006


EXTENSIONS

Corrected typo in third example Dave Zobel (dzobel(AT)alumni.caltech.edu), Mar 07 2009


STATUS

approved



