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A242284
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Greedy residue sequence of triangular numbers 3, 6, 10, 15, ...
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12
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2, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0
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OFFSET
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2,1
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COMMENTS
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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 A242284, s = (1,3,6,10,15,...); s(n) = n(n + 1)/2.
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LINKS
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EXAMPLE
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n .... n(n+1)/2 ... a(n)
1 ... 1 ... (undefined)
2 ... 3 ... 2 = 3 - 1
3 ... 6 ... 2 = 6 - 3 - 1
4 ... 10 .. 0 = 15 - 10 - 3 - 1
5 ... 15 .. 1 = 21 - 15 - 6
6 ... 21 .. 0 = 28 - 21 - 6 - 1
7 ... 28 .. 0 = 36 - 28 - 6 - 1
8 ... 36 .. 1 = 45 - 36 - 6 - 3
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MATHEMATICA
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z = 200; s = Table[n (n + 1)/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}] (* A242284 *)
c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242285 *)
f = 1 + Flatten[Position[tr, 0]] (* A242286 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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