

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



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 *)


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



STATUS

approved



