A257905 Sequence (a(n)) generated by Rule 3 (in Comments) with a(1) = 0 and d(1) = 0.

0, 1, 3, 2, 5, 11, 4, 9, 6, 13, 7, 15, 10, 8, 17, 35, 12, 25, 14, 29, 16, 33, 18, 37, 19, 39, 20, 41, 21, 43, 22, 45, 23, 47, 30, 26, 53, 24, 49, 40, 28, 57, 27, 55, 31, 63, 32, 65, 38, 42, 34, 69, 36, 73, 48, 97, 44, 89, 46, 93, 51, 103, 52, 105, 50, 101

Offset: 1

Author

Clark Kimberling, May 16 2015

Keywords

• nonn
• easy

Comments

Rule 3 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 least 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: suppose that a(1) is an nonnegative integer and d(1) is an integer.

If a(1) = 0 and d(1) != 1, then (a(n)) is a permutation of the nonnegative integers;

if a(1) = 0 and d(1) = 1, then (a(n)) is a permutation of the nonnegative integers excluding 1;

if a(1) = 1, then (a(n)) is a permutation of the positive integers;

if a(1) > 1, then (a(n)) is a permutation of the integers >1;

if d(1) = 0, then (d(n)) is a permutation of the integers;

if d(1) !=0, then (d(n)) is a permutation of the nonzero integers.

Guide to related sequences:

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

0 0 A257905 A258047

0 1 A257906 A257907

0 2 A257908 A257909

0 3 A257910 A257980

1 0 A258046 A258047

1 1 A257981 A257982

1 2 A257983 A257909

2 0 A257985 A257047

2 1 A257986 A257982

2 2 A257987 A257909

Examples

			a(1) = 0, d(1) = 0;
a(2) = 1, d(2) = 1;
a(3) = 3, d(3) = 2;
a(4) = 2, d(4) = -1.
		

Links

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

Crossrefs

Cf. A258047, A257705, A257883, A175498.

Cf. A256283 (putative inverse).

Programs

Haskell

import Data.List ((\\))
a257905 n = a257905_list !! (n-1)
a257905_list = 0 : f [0] [0] where
   f xs@(x:_) ds = g [2 - x .. -1] where
     g [] = y : f (y:xs) (h:ds) where
                  y = x + h
                  (h:_) = [z | z <- [1..] \\ ds, x - z `notElem` xs]
     g (h:hs) | h `notElem` ds && y `notElem` xs = y : f (y:xs) (h:ds)
              | otherwise = g hs
              where y = x + h
-- Reinhard Zumkeller, Jun 03 2015

Mathematica

{a, f} = {{0}, {0}}; Do[tmp = {#, # - Last[a]} &[Min[Complement[#, Intersection[a, #]]&[Last[a] + Complement[#, Intersection[f, #]] &[Range[2 - Last[a], -1]]]]];
If[! IntegerQ[tmp[[1]]], tmp = {Last[a] + #, #} &[NestWhile[# + 1 &, 1, ! (! MemberQ[f, #] && ! MemberQ[a, Last[a] - #]) &]]]; AppendTo[a, tmp[[1]]]; AppendTo[f, tmp[[2]]], {120}]; {a, f} (* Peter J. C. Moses, May 14 2015 *)

Formula

a(n) = A258046(n) - 1 for n >= 1.