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A257018 Rectangular array read by columns: row i shows the numbers whose greedy quarter-squares representation consists of i terms, for i = 1, 2, 3, 4.

%I A257018 #12 Jul 09 2018 23:23:53
%S A257018 0,3,15,255,1,5,19,271,2,7,23,287,4,8,28,304,6,10,33,321,9,11,35,339,
%T A257018 12,13,39,357,16,14,41,376,20,17,45,395,25,18,47,399,30,21,52,415,36,
%U A257018 22,54,419,42,24,59,435,49,26,61,439,56,27,63,456,64,29,67
%N A257018 Rectangular array read by columns: row i shows the numbers whose greedy quarter-squares representation consists of i terms, for i = 1, 2, 3, 4.
%C A257018 Theorem:  Every positive integer is a sum of at most four distinct quarter squares (proved at Math Overflow link).  The greedy representation is found as follows. Let f(n) be the greatest quarter-square <= n, and apply r(n) = f(n) + r(n - f(n)) until reaching 0.  The least term of r(n) is the trace of n, at A257022.
%H A257018 Math Overflow, <a href="http://mathoverflow.net/questions/202903/is-every-positive-integer-a-sum-of-at-most-4-distinct-quarter-squares"> Every positive integer a sum of at most 4 distinct quarter-squares</a>
%e A257018 The array:
%e A257018 0    1    2    4    6    9    12   ...
%e A257018 3    5    7    8    10   11   13   ...
%e A257018 15   19   23   28   33   35   39   ...
%e A257018 255  271  287  304  321  339  357  ...
%e A257018 Quarter-square representations:
%e A257018 r(0) = 0,
%e A257018 r(1) = 1,
%e A257018 r(2) = 2,
%e A257018 r(3) = 2 + 1,
%e A257018 r(15) = 12 + 2 + 1,
%e A257018 r(6969) = 6889 + 72 + 6 + 2.
%t A257018 z = 200; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, z}];
%t A257018 s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
%t A257018 h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[200]; r[0] = {0};
%t A257018 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t A257018 u = Table[Length[r[n]], {n, 0, 4 z}]  (* A257023 *)
%t A257018 TableForm[Table[Take[Flatten[-1 + Position[u, k]], 10], {k, 1, 4}]]  (*A257018 array *)
%t A257018 t = Table[Take[Flatten[-1 + Position[u, k]], 30], {k, 1, 4}];
%t A257018 Flatten[Table[t[[i, j]], {j, 1, 30}, {i, 1, 4}]] (*A257018 sequence *)
%Y A257018 Cf. A257018 (quarter-square sums), A002620 (row 1, the  quarter-squares ), A257019 (row 2), A257020 (row 3); A257021 (row 4), A257023 (number of terms).
%K A257018 nonn,easy,tabf
%O A257018 1,2
%A A257018 _Clark Kimberling_, Apr 15 2015