A064389 Variation (4) on Recamán's sequence (A005132): to get a(n), we first try to subtract n from a(n-1): a(n) = a(n-1) - n if positive and not already in the sequence; if not then we try to add n: a(n) = a(n-1) + n if not already in the sequence; if this fails we try to subtract n+1 from a(n-1), or to add n+1 to a(n-1), or to subtract n+2, or to add n+2, etc., until one of these produces a positive number not already in the sequence - this is a(n).

1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 62, 42, 63, 41, 18, 44, 19, 45, 72, 100, 71, 101, 70, 38, 5, 39, 4, 40, 77, 115, 76, 36, 78, 120, 163, 119, 74, 28, 75, 27, 79, 29, 80, 132, 185, 131, 186, 130, 73, 15, 81, 141, 202

Offset: 1

Author

N. J. A. Sloane, Sep 28 2001

Keywords

• nonn
• easy
• look
• nice

Comments

This is the nicest of these variations. Is this a permutation of the natural numbers?

The number of steps before n appears is the inverse series, A078758. The height of n is in A126712.

See A078758 for the inverse permutation (in case this is a permutation of the positive integers). - M. F. Hasler, Nov 03 2014

After 10^12 terms, the smallest number which has not appeared is 5191516. - Benjamin Chaffin, Oct 09 2016

References

• Suggested by J. C. Lagarias.

Links

• Nick Hobson, Table of n, a(n) for n = 1..10000
• Nick Hobson, Python program for this sequence
• Index entries for sequences related to Recamán's sequence
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A005132, A046901, A064387, A064388. Agrees with A064387 for first 187 terms, then diverges.

Programs

Maple

h := array(1..100000); maxt := 100000; a := array(1..1000); a[1] := 1; h[1] := 1; for nx from 2 to 1000 do for i from 0 to 100 do t1 := a[nx-1]-nx-i; if t1>0 and h[t1] <> 1 then a[nx] := t1; if t1 < maxt then h[t1] := 1; fi; break; fi; t1 := a[nx-1]+nx+i; if h[t1] <> 1 then a[nx] := t1; if t1 < maxt then h[t1] := 1; fi; break; fi; od; od; evalm(a);

Mathematica

h[1] = 1; h[] = 0; maxt = 100000; a[1] = 1; a[] = 0; For[nx = 2, nx <= 1000, nx++, For[i = 0, i <= 100, i++, t1 = a[nx - 1] - nx - i; If[t1 > 0 && h[t1] != 1, a[nx] = t1; If[t1 < maxt, h[t1] = 1]; Break[]]; t1 = a[nx - 1] + nx + i; If[h[t1] != 1, a[nx] = t1; If[t1 < maxt, h[t1] = 1]; Break[]]]]; Table[a[n], {n, 1, 100}](* Jean-François Alcover, May 09 2012, after Maple *)

PARI

A064389(n=1000,show=0)={ my(k,s,t); for(n=1,n, k=n; while( !(t>k && !bittest(s, t-k) && t-=k) && !(!bittest(s, t+k)  && t+=k), k++); s=bitor(s,1<M. F. Hasler, Nov 03 2014