proposed

approved

editing

proposed

allocated for Clark KimberlingSquares-greedy residue of n.

1, 2, 3, 0, 1, 2, 3, 4, 0, 1, 2, 0, 0, 1, 2, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 1, 0, 1, 2, 0, 0, 1, 2, 3, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 1

2,2

Suppose that s = (s(1), s(2), ... ) is a sequence of real numbers such that for every real number u, at most finitely many s(i) are < u, and suppose that x > min(s). We shall apply the greedy algorithm to x, using terms of s. Specifically, let i(1) be an index i such that s(i) = max{s(j) < x}, and put d(1) = x - s(i(1)). If d(1) < s(i) for all i, put r = x - s(i(1)). Otherwise, let i(2) be an index i such that s(i) = max{s(j) < x - s(i(1))}, and put d(2) = x - s(i(1)) - s(i(2)). If d(2) < s(i) for all i, put r = x - s(i(1)) - s(i(2)). Otherwise, let i(3) be an index i such that s(i) = max{s(j) < x - s(i(1)) - s(i(2))}, and put d(3) = x - s(i(1)) - s(i(2)) - s(i(3)). Continue until reaching k such that d(k) < s(i) for every i, and put r = x - s(i(1)) - ... - s(i(k)). Call r the s-greedy residue of x, and call s(i(1)) + ... + s(i(k)) the s-greedy sum for x. If r = 0, call x s-greedy summable.

Clark Kimberling, <a href="/A242305/b242305.txt">Table of n, a(n) for n = 2..2000</a>

n ... a(n)

1 ... (undefined)

2 ... 1 = 2 - 1

3 ... 2 = 3 - 1

4 ... 3 = 4 - 1

5 ... 0 = 5 - 4 - 1

6 ... 1 = 6 - 4 - 1

7 ... 2 = 7 - 4 - 1

8 ... 3 = 8 - 4 - 1

9 ... 4 = 9 - 4 - 1

10 .. 0 = 10 - 9 - 1

11 .. 1 = 11 - 9 - 1

12 .. 2 = 12 - 9 - 1

13 .. 0 = 13 - 9 - 4

z = 200; s = Table[n^2, {n, 1, z}]; s1 = Table[n, {n, 1, z}]; t = Table[{s1[[n]], #, Total[#] == s1[[n]]} &[DeleteCases[-Differences[FoldList[If[#1 - #2 >= 0, #1 - #2, #1] &, s1[[n]],

Reverse[Select[s, # < s1[[n]] &]]]], 0]], {n, z}]

r[n_] := s1[[n]] - Total[t[[n]][[2]]];

tr = Table[r[n], {n, 2, z}] (* A242305 *)

c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242306 *)

f = 1 + Flatten[Position[tr, 0]] (* A242307 *) (* Peter J. C. Moses, May 06 2014 *)

Cf. A242306, A242307, A241833, A000027, A000290.

allocated

nonn,easy

Clark Kimberling, May 11 2014

approved

editing

allocating

allocated

allocated for Clark Kimberling

allocating

approved