A078783 a(0) = 0; a(1)=1; for n>1, a(n) = least positive integer m not among a(1),...,a(n-1) such that |m-a(n-1)| > |a(n-1)-a(n-2)|.

0, 1, 3, 6, 2, 7, 13, 4, 14, 25, 5, 26, 48, 8, 49, 91, 9, 92, 176, 10, 177, 345, 11, 346, 682, 12, 683, 1355, 15, 1356, 2698, 16, 2699, 5383, 17, 5384, 10752, 18, 10753, 21489, 19, 21490, 42962, 20, 42963, 85907, 21, 85908, 171796, 22, 171797

Offset: 0

Author

Reiner Martin, Jan 09 2003

Keywords

• easy
• nonn

Comments

This is a permutation pi of the nonnegative integers such that |pi(n+1)-pi(n)| is strictly increasing. In other words, it is a walk on the nonnegative numbers with strictly increasing step size which visits every number exactly once.

A greedy version of Recamán's sequence: Construct two sequences a() and d() as follows. a(0)=0, a(1)=1, a(2)=3, d(0)=0, d(1)=1, d(2)=2. For n>=3, let m be smallest nonnegative number not yet in the a sequence. Let i = a(n-1)-m. If i > d(n), then a(n) = a(n-1)-i = m, d(n) = i; otherwise a(n) = a(n-1)+d(n-1)+1, d(n) = d(n-1)+1. Has the properties that a() is the Recamán transform of d() and every number appears in a(). This sequence is a(), while d() is A117073. Has a natural decompostion into segments of length 3. - N. J. A. Sloane, Apr 16 2006

For n>0: a(3*n-2)=A117070(n), a(3*n-1)=A117071(n) and a(3*n)=A117072(n).

References

• N. J. A. Sloane and Allan Wilks, On sequences of Recaman type, paper in preparation, 2006.

Links

• Paul Tek, Table of n, a(n) for n = 0..2000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A072007, A005132, A117070-A117075.

Cf. A257502 (inverse).

Programs

Haskell

import Data.List (delete)
a078783 n = a078783_list !! n
(a078783_list, a117073_list) = unzip $
   (0,0) : (1,1) : (3,2) : f 3 2 (2:[4..]) where
   f a d ms@(m:_) = (a', d') : f a' d' (delete a' ms) where
     (a', d') = if i > d then (m, i) else (a + d + 1, d + 1)
     i = a - m
-- Reinhard Zumkeller, May 01 2015

Mathematica

a[0] = 0; a[1] = 1;
a[n_] := a[n] = For[m = 2, True, m++, If[FreeQ[Array[a, n-1], m], If[Abs[m - a[n-1]] > Abs[a[n-1] - a[n-2]], Return[m]]]];
Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 02 2018 *)