

A257878


Sequence (a(n)) generated by Rule 1 (in Comments) with a(1) = 1 and d(1) = 1.


2



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, 34
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OFFSET

1,2


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.
Considering the first 1000 elements of this sequence and A257705 it appears that this is the same as A257705 apart from an index shift.  R. J. Mathar, May 14 2015


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) = 1, d(1) = 1;
a(2) = 3, d(2) = 2;
a(3) = 2, d(3) = 1;
a(4) = 5, d(4) = 3.


MATHEMATICA

a[1] = 1; d[1] = 1; 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}] (* A257878 *)
Table[d[k], {k, 1, zz}] (* A131389 essentially *)


CROSSREFS

Cf. A131388, A131389, A257705, A081145, A257883, A175498.
Sequence in context: A320274 A333398 A257705 * A243700 A193796 A249906
Adjacent sequences: A257875 A257876 A257877 * A257879 A257880 A257881


KEYWORD

nonn,easy


AUTHOR

Clark Kimberling, May 12 2015


STATUS

approved



