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A242288 Greedy residue sequence of tetrahedral numbers 4, 10, 20, 35, ... 5
3, 5, 5, 0, 0, 3, 0, 0, 0, 0, 1, 2, 0, 0, 1, 2, 1, 0, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 4, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 2, 0, 0, 0, 1, 0, 0, 4, 0, 2, 0, 3, 0, 2, 4 (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 A242288, s = (1,4,10,20,...); s(n) = n(n + 1)(n+2)/6.

LINKS

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

EXAMPLE

n ... n(n+1)(n+2)/6 ... a(n)

1 ..  1 ............... (undefined)

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

3 ... 10 .............. 5 = 10 - 4 - 1

4 ... 20 .............. 15 = 20 - 10 - 4 - 1

5 ... 35 .............. 0 = 35 - 20 - 10 - 4 -1 1

6 ... 56 .............. 0 = 56 - 35 - 20 - 1

7 ... 84 .............. 3 = 84 - 56 - 20 - 4 - 1

8 ... 120 ............. 0 = 120 - 84 - 35 - 1

9 ... 165 ............. 0 = 165 - 120 - 35 - 10

10 .. 220 ............. 0 = 220 - 165 - 35 - 20

11 .. 286 ............. 0 = 286 - 220 - 56 - 10

MATHEMATICA

z = 200;  s = Table[n (n + 1)(n + 2)/6, {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}]  (* A242288 *)

c = Table[Length[t[[n]][[2]]], {n, 2, z}] (* A242289 *)

f = 1 + Flatten[Position[tr, 0]]  (* A242290*)

f (f + 1)(f + 2)/6  (* A242291 *) (* Peter J. C. Moses, May 06 2014 *)

CROSSREFS

Cf. A242289, A242290, A242291, A241833, A242284, A000292.

Sequence in context: A161353 A182045 A133758 * A021742 A284867 A152416

Adjacent sequences:  A242285 A242286 A242287 * A242289 A242290 A242291

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 10 2014

STATUS

approved

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Last modified June 18 14:52 EDT 2019. Contains 324213 sequences. (Running on oeis4.)