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A257705 Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 0 and d(1) = 0. 20
0, 1, 3, 2, 5, 9, 7, 4, 10, 6, 11, 18, 13, 21, 15, 8, 17, 27, 19, 30, 20, 32, 23, 12, 25, 39, 26, 14, 29, 45, 31, 16, 33, 51, 35, 54, 37, 57, 38, 59, 41, 63, 43, 22, 46, 24, 47, 72, 49, 75, 50, 77, 53, 81, 55, 28, 58, 87, 56, 88, 60, 91, 62, 95, 65, 99, 67 (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.

Guide to related sequences:

a(1)  d(1)      (a(n))             (d(n))

0       0      A257705      A131389 except for initial terms

0       1      A257706      A131389 except for initial terms

0       2      A257876      A131389 except for initial terms

0       3      A257877      A257915

1       0      A131388      A131389

1       1      A257878      A131389 except for initial terms

2       0      A257879      A257880

2       1      A257881      A257880 except for initial terms

2       2      A257882      A257918

LINKS

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

FORMULA

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

Also, a(k) = A131388(n)-1.

EXAMPLE

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

a(3) = a(2) + d(3) = 1 + 2 = 3;

a(4) = a(3) + d(4) = 3 + (-1) = 2.

MATHEMATICA

a[1] = 0; d[1] = 0; 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}] (* A257705 *)

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

CROSSREFS

Cf. A131388, A081145, A257883, A175498.

Sequence in context: A254331 A210742 A175056 * A257878 A243700 A193796

Adjacent sequences:  A257702 A257703 A257704 * A257706 A257707 A257708

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 12 2015

STATUS

approved

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Last modified May 23 19:46 EDT 2017. Contains 286926 sequences.