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A257876 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 2. 2
0, 1, 4, 3, 7, 5, 2, 8, 13, 9, 16, 11, 19, 12, 6, 15, 25, 17, 28, 18, 30, 21, 10, 23, 37, 24, 39, 27, 43, 29, 14, 31, 49, 33, 52, 35, 55, 36, 57, 34, 56, 38, 61, 41, 20, 44, 22, 47, 73, 48, 75, 51, 79, 53, 26, 58, 87, 59, 89, 60, 91, 54, 88, 50, 83, 42, 77 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Rule 1 follows.  For k >= 1, let  A(k) = {a(1), …, a(k)} and D(k) = {d(1), …, d(k)}.  Begin with k = 1 and nonnegative integers a(1) and d(1).

Step 1:   If there is an integer h such that 1 - a(k) < h < 0 and h is not in D(k) and a(k) + h is not in A(k), let d(k+1) be the greatest such h, let a(k+1) = a(k) + h, replace k by k + 1, and repeat Step 1; otherwise do Step 2.

Step 2:  Let h be the least positive integer not in D(k) such that a(k) + h is not in A(k).  Let a(k+1) = a(k) + h and d(k+1) = h.  Replace k by k+1 and do Step 1.

Conjecture:  if a(1) is an nonnegative integer and d(1) is an integer, then (a(n)) is a permutation of the nonnegative integers (if a(1) = 0) or a permutation of the positive integers (if a(1) > 0).  Moreover, (d(n)) is a permutation of the integers if d(1) = 0, or of the nonzero integers if d(1) > 0.

See A257705 for a guide to related sequences.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000

FORMULA

a(k+1) - a(k) = d(k+1) for k >= 1.

EXAMPLE

a(1) = 0, d(1) = 2;

a(2) = 1, d(2) = 1;

a(3) = 3, d(3) = 3;

a(4) = 4, d(4) = -1.

The first terms of (d(n)) are (2,1,3,-1,4,-2,-3,6,5,...), which differs from A131389 only in initial terms.

MATHEMATICA

a[1] = 0; d[1] = 2; k = 1; z = 10000; zz = 120;

A[k_] := Table[a[i], {i, 1, k}]; diff[k_] := Table[d[i], {i, 1, k}];

c[k_] := Complement[Range[-z, z], diff[k]];

T[k_] := -a[k] + Complement[Range[z], A[k]];

s[k_] := Intersection[Range[-a[k], -1], c[k], T[k]];

Table[If[Length[s[k]] == 0, {h = Min[Intersection[c[k], T[k]]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}, {h = Max[s[k]], a[k + 1] = a[k] + h, d[k + 1] = h, k = k + 1}], {i, 1, zz}];

u = Table[a[k], {k, 1, zz}]  (* A257876 *)

Table[d[k], {k, 1, zz}]      (* A131389 essentially *)

CROSSREFS

Cf. A131388, A257705, A081145, A257883, A175498.

Sequence in context: A127752 A198874 A276760 * A093051 A089020 A046560

Adjacent sequences:  A257873 A257874 A257875 * A257877 A257878 A257879

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 12 2015

STATUS

approved

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Last modified July 19 08:33 EDT 2019. Contains 325155 sequences. (Running on oeis4.)