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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

LINKS

Table of n, a(n) for n=1..63.

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

EXAMPLE

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.

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 *)

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).

Sequence in context: A288987 A288986 A288988 * A173146 A139289 A250405

Adjacent sequences:  A257015 A257016 A257017 * A257019 A257020 A257021

KEYWORD

nonn,easy,tabl

AUTHOR

Clark Kimberling, Apr 15 2015

STATUS

approved

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Last modified August 17 06:10 EDT 2017. Contains 290635 sequences.