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A294524 Numbers that have a unique partition into a sum of five nonnegative squares.

%I A294524 #27 Feb 25 2019 09:32:57
%S A294524 0,1,2,3,6,7,15
%N A294524 Numbers that have a unique partition into a sum of five nonnegative squares.
%C A294524 This sequence is finite and complete. See the von Eitzen Link and the proof in A294675 stating that for n > 5408, the number of ways to write n as a sum of 5 squares (without allowing zero squares) is at least floor(sqrt(n - 101) / 8) = 9. Since this sequence relaxes the restriction of zero squares the number of representations for n > 5408 is at least nine. Then an inspection of n <= 5408 completes the proof.
%D A294524 E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, New York, 1985, p. 86, Theorem 1.
%H A294524 H. von Eitzen, in reply to user James47, <a href="http://math.stackexchange.com/questions/811824/what-is-the-largest-integer-with-only-one-representation-as-a-sum-of-five-nonzer">What is the largest integer with only one representation as a sum of five nonzero squares?</a> on stackexchange.com, May 2014
%H A294524 D. H. Lehmer, <a href="http://www.jstor.org/stable/2305380">On the Partition of Numbers into Squares</a>, The American Mathematical Monthly, Vol. 55, No. 8, October 1948, pp. 476-481.
%t A294524 m = 5;
%t A294524 r[n_] := Reduce[xx = Array[x, m]; 0 <= x[1] && LessEqual @@ xx && AllTrue[xx, NonNegative] && n == Total[xx^2], xx, Integers];
%t A294524 For[n = 0, n < 20, n++, rn = r[n]; If[rn[[0]] === And, Print[n, " ", rn]]] (* _Jean-François Alcover_, Feb 25 2019 *)
%Y A294524 Cf. A000174, A006431, A294675.
%K A294524 nonn,fini,full
%O A294524 1,3
%A A294524 _Robert Price_, Nov 01 2017