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%I #12 Jul 09 2018 23:23:53
%S 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 12,13,39,357,16,14,41,376,20,17,45,395,25,18,47,399,30,21,52,415,36,
%U 22,54,419,42,24,59,435,49,26,61,439,56,27,63,456,64,29,67
%N 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 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 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 The array:
%e 0 1 2 4 6 9 12 ...
%e 3 5 7 8 10 11 13 ...
%e 15 19 23 28 33 35 39 ...
%e 255 271 287 304 321 339 357 ...
%e Quarter-square representations:
%e r(0) = 0,
%e r(1) = 1,
%e r(2) = 2,
%e r(3) = 2 + 1,
%e r(15) = 12 + 2 + 1,
%e r(6969) = 6889 + 72 + 6 + 2.
%t z = 200; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, z}];
%t s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
%t h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[200]; r[0] = {0};
%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t u = Table[Length[r[n]], {n, 0, 4 z}] (* A257023 *)
%t TableForm[Table[Take[Flatten[-1 + Position[u, k]], 10], {k, 1, 4}]] (*A257018 array *)
%t t = Table[Take[Flatten[-1 + Position[u, k]], 30], {k, 1, 4}];
%t Flatten[Table[t[[i, j]], {j, 1, 30}, {i, 1, 4}]] (*A257018 sequence *)
%Y 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 nonn,easy,tabf
%O 1,2
%A _Clark Kimberling_, Apr 15 2015
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