

A242305


Squaresgreedy residue of n.


3



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
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OFFSET

2,2


COMMENTS

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 sgreedy residue of x, and call s(i(1)) + ... + s(i(k)) the sgreedy sum for x. If r = 0, call x sgreedy summable.


LINKS

Clark Kimberling, Table of n, a(n) for n = 2..2000


EXAMPLE

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


MATHEMATICA

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 *)


CROSSREFS

Cf. A242306, A242307, A241833, A000027, A000290.
Sequence in context: A277904 A132387 A124757 * A049263 A014588 A293497
Adjacent sequences: A242302 A242303 A242304 * A242306 A242307 A242308


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 11 2014


STATUS

approved



