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A063772 a(k^2 + i) = k + a(i) for k >= 0 and 0 <= i <= k * 2; a(0) = 0.

%I A063772 #20 Mar 04 2019 09:21:14
%S A063772 0,1,2,3,2,3,4,5,4,3,4,5,6,5,6,7,4,5,6,7,6,7,8,9,8,5,6,7,8,7,8,9,10,9,
%T A063772 8,9,6,7,8,9,8,9,10,11,10,9,10,11,12,7,8,9,10,9,10,11,12,11,10,11,12,
%U A063772 13,12,13,8,9,10,11,10,11,12,13,12,11,12,13,14,13,14,15,12,9,10,11,12
%N A063772 a(k^2 + i) = k + a(i) for k >= 0 and 0 <= i <= k * 2; a(0) = 0.
%C A063772 a(n^2) = n (by definition).
%C A063772 First difference from A138554 is a(32) = 10 (5^2+2^2+1^2+1^2+1^2), A138554(32) = 8 (4^2+4^2). - _Franklin T. Adams-Watters_, Jun 25 2015
%C A063772 a(n) is the sum of the roots of the summands when n is expressed as a sum of squares using the greedy algorithm (as in A053610). - _Franklin T. Adams-Watters_, Jun 30 2015
%H A063772 Robert Israel, <a href="/A063772/b063772.txt">Table of n, a(n) for n = 0..10000</a>
%p A063772 a:= proc(n)  option remember;
%p A063772       local k;
%p A063772       k:= floor(sqrt(n));
%p A063772       k + procname(n-k^2);
%p A063772 end proc:
%p A063772 a(0):= 0:
%p A063772 map(a, [$0..100]); # _Robert Israel_, Jun 30 2015
%t A063772 a[0] = 0; a[n_] := a[n] = With[{k = n // Sqrt // Floor}, k + a[n-k^2]];
%t A063772 Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Mar 04 2019 *)
%Y A063772 Cf. A053610, A138554.
%K A063772 nonn,look
%O A063772 0,3
%A A063772 _Reinhard Zumkeller_, Aug 15 2001