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%I #9 Jun 13 2016 08:21:40
%S 1,5,10,10,10,21,30,20,15,35,50,40,30,45,70,60,30,55,100,80,56,90,110,
%T 80,60,85,120,100,60,90,130,80,35,90,120,80,65,85,90,60,35,60,90,50,
%U 30,61,60,20,10,50,40,30,25,20,30,0,10,20,20,10,0,20,0,0,5,5,10,0,5,0,0,0,0,5,0,0,0,0,0,0,1
%N Number of ordered ways to write n as the sum of 5 squares less than 5^2.
%C Through n = 24, a(n) = number of ordered ways to write n as the sum of 5 squares. For n > 24, we must exclude sums which include 5^2, 6^2 and the like. The values of n such that a(n) = 0 are 55, 60, 62, 63, 67, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79 and all n > 80. 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.
%e a(0) = 1 because the unique such sum is 0 = 0^2 + 0^2 + 0^2 + 0^2 + 0^2.
%e a(1) = 5 because there are 5 permutations of 1 = 1^2 + 0^2 + 0^2 + 0^2 + 0^2, such as 1 = 0^2 + 1^2 + 0^2 + 0^2 + 0^2.
%e a(2) = 10 because there are 10 permutations of 2 = 1^2 + 1^2 + 0^2 + 0^2 + 0^2, such as 2 = 1^2 + 0^2 + 1^2 + 0^2 + 0^2.
%e a(5) = 21 because of the unique sum 5 = 1^2 + 1^2 + 1^2 + 1^2 + 1^2 and also 20 permutations of 5 = 2^2 + 1^2 + 0^2 + 0^2 + 0^2.
%e a(16) = 30 because there are 5 permutations of 16 = 4^2 + 0^2 + 0^2 + 0^2 + 0^2 and 5 permutations of 16 = 0^2 + 2^2 + 2^2 + 2^2 + 2^2 and 20 permutations of 16 = 3^2 + 2^2 + 1^2 + 1^2 + 1^2.
%t a[n_] := Total[ Length /@ Permutations /@ IntegerPartitions[n, {5}, Range[0, 4]^2]]; a /@ Range[0, 80] (* _Giovanni Resta_, Jun 13 2016 *)
%Y Cf. A000118, A014110.
%K easy,fini,full,nonn
%O 0,2
%A _Jonathan Vos Post_, Oct 11 2006
%E 23 terms corrected by _Giovanni Resta_, Jun 13 2016
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