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 A242284 Greedy residue sequence of triangular numbers 3, 6, 10, 15, ... 12
 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 (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 A242284, s = (1,3,6,10,15,...); s(n) = n(n + 1)/2. LINKS Clark Kimberling, Table of n, a(n) for n = 2..2000 EXAMPLE 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 MATHEMATICA 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]][]]; tr = Table[r[n], {n, 2, z}]  (* A242284 *) c = Table[Length[t[[n]][]], {n, 2, z}] (* A242285 *) f = 1 + Flatten[Position[tr, 0]]  (* A242286 *) f (f + 1)/2  (* A242287 *) (* Peter J. C. Moses, May 06 2014 *) CROSSREFS Cf. A242285, A242286, A242287, A241833, A242288, A000217. Sequence in context: A116511 A248211 A049502 * A333624 A306595 A332996 Adjacent sequences:  A242281 A242282 A242283 * A242285 A242286 A242287 KEYWORD nonn,easy AUTHOR Clark Kimberling, May 10 2014 STATUS approved

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Last modified July 10 07:08 EDT 2020. Contains 335573 sequences. (Running on oeis4.)