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.

0, 3, 15, 255, 1, 5, 19, 271, 2, 7, 23, 287, 4, 8, 28, 304, 6, 10, 33, 321, 9, 11, 35, 339, 12, 13, 39, 357, 16, 14, 41, 376, 20, 17, 45, 395, 25, 18, 47, 399, 30, 21, 52, 415, 36, 22, 54, 419, 42, 24, 59, 435, 49, 26, 61, 439, 56, 27, 63, 456, 64, 29, 67

Offset: 1

Author

Clark Kimberling, Apr 15 2015

Keywords

• nonn
• easy
• tabf

Comments

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.

Examples

			The array:
0    1    2    4    6    9    12   ...
3    5    7    8    10   11   13   ...
15   19   23   28   33   35   39   ...
255  271  287  304  321  339  357  ...
Quarter-square representations:
r(0) = 0,
r(1) = 1,
r(2) = 2,
r(3) = 2 + 1,
r(15) = 12 + 2 + 1,
r(6969) = 6889 + 72 + 6 + 2.
		

Links

• Math Overflow, Every positive integer a sum of at most 4 distinct quarter-squares

Crossrefs

Cf. A257018 (quarter-square sums), A002620 (row 1, the quarter-squares ), A257019 (row 2), A257020 (row 3); A257021 (row 4), A257023 (number of terms).

Programs

Mathematica

z = 200; b[n_] := Floor[(n + 1)^2/4]; bb = Table[b[n], {n, 0, z}];
s[n_] := Table[b[n], {k, b[n + 1] - b[n]}];
h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
u = Table[Length[r[n]], {n, 0, 4 z}]  (* A257023 *)
TableForm[Table[Take[Flatten[-1 + Position[u, k]], 10], {k, 1, 4}]]  (*A257018 array *)
t = Table[Take[Flatten[-1 + Position[u, k]], 30], {k, 1, 4}];
Flatten[Table[t[[i, j]], {j, 1, 30}, {i, 1, 4}]] (*A257018 sequence *)