0,3

The name "Recamán's sequence" is due to N. J. A. Sloane, not the author!

I conjecture that every number eventually appears - see A057167, A064227, A064228. - N. J. A. Sloane. That was written in 1991. Today I'm not so sure that every number appears. - N. J. A. Sloane, Feb 26 2017

From David W. Wilson, Jul 13 2009: (Start)

The sequence satisfies [1] a(n) >= 0, [2] |a(n)-a(n-1)| = n, and tries to avoid repeats by greedy choice of a(n) = a(n-1) -+ n.

This "wants" to be an injection on N = {0, 1, 2, ...}, as it attempts to avoid repeats by choice of a(n) = a(n-1) + n when a(n) = a(n-1) - n is a repeat.

Clearly, there are injections satisfying [1] and [2], e.g., a(n) = n(n+1)/2.

Is there a lexicographically earliest injection satisfying [1] and [2]? (End)

The definition can be written as a(n) = a(n-1) - n*(-1)^(c(n)+d(n)) where c(n) and d(n) are two flags defined via b(n) = a(n-1)-n; theta(n) = 1 if n>0, =0 if n<=0; c(n) = theta(1+b(n)); d(n) = theta( product_{j=0..n-1}( a(j)-b(n))^2 ). - Adriano Caroli, Dec 26 2010

It appears that a(n) is also the value of "x" and "y" of the endpoint of the L-toothpick structure mentioned in A210606 after n-th stage. - Omar E. Pol, Mar 24 2012

Of course this is not a permutation of the integers: the first repeated term is 42 = a(24) = a(20). - M. F. Hasler, Nov 03 2014. Also 43 = a(18) = a(26). - Jon Perry, Nov 06 2014

Bernardo Recamán Santos, letter to N. J. A. Sloane, Jan 29 1991

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

N. J. A. Sloane, The first 100000 terms

Benjamin Chaffin, Log-log plot of first 10^230 terms

GBnums, See: A nice OEIS sequence

Gordon Hamilton, Wrecker Ball Sequences, Video, 2013

Nick Hobson, Python program for this sequence

C. L. Mallows, Plot (jpeg) of first 10000 terms

C. L. Mallows, Plot (postscript) of first 10000 terms

Tilman Piesk, First 172 items in a coordinate system [This is a graph of the start of A005132 rotated clockwise by 90 degs. - N. J. A. Sloane, Mar 23 2012]

Omar E. Pol, Illustration of initial terms

Omar E. Pol, Illustration of initial terms of A001057, A005132, A000217

N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228

Alex van den Brandhof, Een bizarre rij, Pythagoras, 55ste Jaargang, Nummer 2, Nov 2015, (Article in Dutch about this sequence, see page 19 and back cover).

Index entries for sequences related to Recamán's sequence

Consider n=6. We have a(5)=7 and try to subtract 6. The result, 1, is certainly positive, but we cannot use it because 1 is already in the sequence. So we must add 6 instead, getting a(6) = 7 + 6 = 13.

h := array(1..100000); maxt := 100000; a := [1]; ad := [1]; su := []; h[1] := 1; for nx from 2 to 500 do t1 := a[nx-1]-nx; if t1>0 and h[t1] <> 1 then su := [op(su), nx]; else t1 := a[nx-1]+nx; ad := [op(ad), nx]; fi; a := [op(a), t1]; if t1 <= maxt then h[t1] := 1; fi; od: # a is A005132, ad is A057165, su is A057166

A005132 := proc(n)

        option remember;

        local a, fou, j ;

        if n = 0 then 0;

        else

                a := procname(n-1)-n ;

                if a <= 0 then return a+2*n ; else fou := false; for j from 0 to n-1 do if procname(j) = a then fou :=true; break; end if; end do; if fou then a+2*n; else a ; end if;

                end if;

        end if;

end proc: # R. J. Mathar, Apr 01 2012

a = {1}; Do[ If[ a[ [ -1 ] ] - n > 0 && Position[ a, a[ [ -1 ] ] - n ] == {}, a = Append[ a, a[ [ -1 ] ] - n ], a = Append[ a, a[ [ -1 ] ] + n ] ], {n, 2, 70} ]; a

f[s_List] := Block[{a = s[[ -1]], len = Length@s}, Append[s, If[a > len && !MemberQ[s, a - len], a - len, a + len]]]; Nest[f, {0}, 70] (* Robert G. Wilson v, May 01 2009 *)

(PARI) a(n)=if(n<2, 1, if(abs(sign(a(n-1)-n)-1)+setsearch(Set(vector(n-1, i, a(i))), a(n-1)-n), a(n-1)+n, a(n-1)-n))  \\ Benoit Cloitre

(PARI) A005132(N=1000, show=0)={ my(s, t); for(n=1, N, s=bitor(s, 1<<t += if( t<=n || bittest(s, t-n), n, -n)); show&&print1(t", ")); t} \\ M. F. Hasler, May 11 2008, updated M. F. Hasler, Nov 03 2014

(MATLAB)

function a=A005132(m)

% m=max number of terms in a(n). Offset n:0

a=zeros(1, m);

for n=2:m

    B=a(n-1)-(n-1);

    C=0.^( abs(B+1) + (B+1) );

    D=ismember(B, a(1:n-1));

    a(n)=a(n-1)+ (n-1) * (-1)^(C + D -1);

end

% Adriano Caroli, Dec 26 2010

(Haskell)

import Data.Set (Set, singleton, notMember, insert)

a005132 n = a005132_list !! n

a005132_list = 0 : recaman (singleton 0) 1 0 where

   recaman :: Set Integer -> Integer -> Integer -> [Integer]

   recaman s n x = if x > n && (x - n) `notMember` s

                      then (x-n) : recaman (insert (x-n) s) (n+1) (x-n)

                      else (x+n) : recaman (insert (x+n) s) (n+1) (x+n)

-- Reinhard Zumkeller, Mar 14 2011

Cf. A057165 (addition steps), A057166 (subtraction steps), A057167 (steps to hit n), A008336, A046901 (simplified version), A064227 (records for reaching n), A064228 (value of n that take a record number of steps to reach), A064284 (no. of times n appears), A064290 (heights of terms), A064291 (record highs), A119632 (condensed version), A063733, A079053, A064288, A064289, A064387, A064388, A064389, A228474 (bidirectional version).

A065056 gives partial sums, A160356 gives first differences.

A row of A066201.

Sequence in context: A076543 A274648 A277558 * A064388 A064387 A064389

Adjacent sequences:  A005129 A005130 A005131 * A005133 A005134 A005135

nonn,nice,hear,look

N. J. A. Sloane and Simon Plouffe, May 16 1991

Allan Wilks, Nov 06 2001, computed 10^15 terms of this sequence. At this point the smallest missing number was 852655 = 5*31*5501.

After 10^25 terms of A005132 the smallest missing number is still 852655. - Benjamin Chaffin, Jun 13 2006

Even after 7.78*10^37 terms, the smallest missing number is still 852655. - Benjamin Chaffin, Mar 28 2008

Even after 4.28*10^73 terms, the smallest missing number is still 852655. - Benjamin Chaffin, Mar 22 2010

Even after 10^230 terms, the smallest missing number is still 852655. - Benjamin Chaffin, Mar 22 2010

approved