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A242305 Squares-greedy 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 (list; graph; refs; listen; history; text; internal format)
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 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.

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

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Last modified January 16 21:37 EST 2019. Contains 319206 sequences. (Running on oeis4.)