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A241833 Greedy residue sequence of squares 2^2, 3^2, 4^2, ... 26
3, 4, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 3, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 3, 0, 2, 0, 0, 2, 0, 2, 0, 1, 0, 1, 0, 1, 0, 1, 3, 0, 2, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 3, 0, 2, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

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.  If s(1) = min(s) < s(2), then taking x = s(i) successively for i = 2, 3,... gives a residue r(i) for each i; call (r(i)) the greedy residue sequence for s.  When s is understood from context, the prefix "s-" is omitted.  For A241833, s = (1^2, 2^2, 3^2, 4^2, ... ).

LINKS

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

EXAMPLE

n ... n^2 .. a(n)

1 ... 1 .... (undefined)

2 ... 4 .... 3 = 4 - 1

3 ... 9 .... 4 = 9 - 4 - 1

4 ... 16 ... 2 = 16 - 9 - 4 - 1

5 ... 25 ... 0 = 25 - 16 - 9

6 ... 36 ... 1 = 36 - 25 - 9 - 1

7 ... 49 ... 0 = 49 - 36 - 9 - 4

8 ... 64 ... 1 = 64 - 49 - 9 - 4 - 1

MATHEMATICA

z = 200;  s = Table[n^2, {n, 1, z}]; t = Table[{s[[n]], #, Total[#] == s[[n]]} &[   DeleteCases[-Differences[FoldList[If[#1 - #2 >= 0, #1 - #2, #1] &, s[[n]], Reverse[Select[s, # < s[[n]] &]]]], 0]], {n, z}]; r[n_] := s[[n]] - Total[t[[n]][[2]]]; tr =  Table[r[n], {n, 2, z}]  (* A241833 *)

c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A241834 *)

f = 1 + Flatten[Position[tr, 0]]  (* A241835 *)

f^2  (* A241836 *)

(* Peter J. C. Moses, May 06 2014 *)

CROSSREFS

Cf. A241834, A241835, A241836, A242252, A000290.

Sequence in context: A123951 A123127 A167876 * A077451 A019829 A200125

Adjacent sequences:  A241830 A241831 A241832 * A241834 A241835 A241836

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 09 2014

STATUS

approved

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Last modified August 17 06:03 EDT 2017. Contains 290635 sequences.