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A257918 Sequence (d(n)) generated by Rule 1 (in Comments) with a(1) = 2 and d(1) = 2. 3
2, -1, 3, 1, -2, 4, 5, -3, 6, -4, -5, 7, 8, -7, -6, 9, 10, -8, -9, 12, 11, -10, 13, -11, 14, -13, 15, -12, 16, -14, -15, 18, -16, 17, 19, -17, 20, -19, -18, 22, 21, -20, 23, -22, 24, -21, -23, 25, 26, -24, 27, -25, 28, -26, -27, 30, -28, 29, 31, -29, 32, -31 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

This is the sequence (d(n)) of differences associated with the sequence a = A257882.

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

d(k) = a(k) - a(k-1) for k >=2, where a(k) = A257882(k).

EXAMPLE

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

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

a(3) = 4, d(3) = 3;

a(4) = 5, d(4) = 1.

MATHEMATICA

a[1] = 2; 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}]  (* A257882 *)

Table[d[k], {k, 1, zz}]      (* A257918 *)

CROSSREFS

Cf. A131389, A257705, A081145, A257918, A175499.

Sequence in context: A270656 A045898 A303754 * A257912 A036262 A080521

Adjacent sequences:  A257915 A257916 A257917 * A257919 A257920 A257921

KEYWORD

easy,sign

AUTHOR

Clark Kimberling, May 13 2015

STATUS

approved

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Last modified January 22 22:16 EST 2020. Contains 331166 sequences. (Running on oeis4.)