

A307531


a(n) is the greatest sum i + j + k + l where i^2 + j^2 + k^2 + l^2 = n and 0 <= i <= j <= k <= l.


2



0, 1, 2, 3, 4, 3, 4, 5, 4, 5, 6, 5, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 8, 9, 8, 9, 10, 9, 10, 9, 10, 11, 8, 11, 10, 11, 12, 11, 12, 11, 12, 11, 12, 13, 12, 13, 12, 13, 12, 13, 14, 13, 14, 13, 14, 13, 12, 15, 14, 15, 14, 15, 14, 15, 16, 15, 16, 15, 16, 15, 16, 15
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OFFSET

0,3


COMMENTS

The sequence is well defined as every nonnegative integer can be represented as a sum of four squares in at least one way.
It appears that a(n^2) = 2*n if n is even and 2*n1 if n is odd.  Robert Israel, Apr 14 2019


LINKS

Robert Israel, Table of n, a(n) for n = 0..10000
Rémy Sigrist, C program for A307531
Wikipedia, Lagrange's foursquare theorem


EXAMPLE

For n = 34:
 34 can be expressed in 4 ways as a sum of four squares:
i^2 + j^2 + k^2 + l^2 i+j+k+l
 
0^2 + 0^2 + 3^2 + 5^2 8
0^2 + 3^2 + 3^2 + 4^2 10
1^2 + 1^2 + 4^2 + 4^2 10
1^2 + 2^2 + 2^2 + 5^2 10
 a(34) = max(8, 10) = 10.


MAPLE

g:= proc(n, k) option remember; local a;
if k = 1 then if issqr(n) then return sqrt(n) else return infinity fi fi;
max(seq(a+procname(na^2, k1), a=0..floor(sqrt(n))))
end proc:
seq(g(n, 4), n=0..100); # Robert Israel, Apr 14 2019


MATHEMATICA

Array[Max[Total /@ PowersRepresentations[#, 4, 2]] &, 68, 0] (* Michael De Vlieger, Apr 13 2019 *)


PROG

(C) See Links section.


CROSSREFS

See A307510 for the multiplicative variant.
Cf. A002635.
Sequence in context: A114524 A058033 A216197 * A125619 A262519 A225320
Adjacent sequences: A307528 A307529 A307530 * A307532 A307533 A307534


KEYWORD

nonn


AUTHOR

Rémy Sigrist, Apr 13 2019


STATUS

approved



