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%I M2511 #357 Oct 22 2023 22:59:02
%S 0,1,3,6,2,7,13,20,12,21,11,22,10,23,9,24,8,25,43,62,42,63,41,18,42,
%T 17,43,16,44,15,45,14,46,79,113,78,114,77,39,78,38,79,37,80,36,81,35,
%U 82,34,83,33,84,32,85,31,86,30,87,29,88,28,89,27,90,26,91,157,224,156,225,155
%N Recamán's sequence (or Recaman's sequence): a(0) = 0; for n > 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence, otherwise a(n) = a(n-1) + n.
%C The name "Recamán's sequence" is due to _N. J. A. Sloane_, not the author!
%C 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
%C As of Jan 25 2018, the first 13 missing numbers are 852655, 930058, 930557, 964420, 966052, 966727, 969194, 971330, 971626, 971866, 972275, 972827, 976367, ... For further information see the "Status Report" link. - _Benjamin Chaffin_, Jan 25 2018
%C From _David W. Wilson_, Jul 13 2009: (Start)
%C 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.
%C 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.
%C Clearly, there are injections satisfying [1] and [2], e.g., a(n) = n(n+1)/2.
%C Is there a lexicographically earliest injection satisfying [1] and [2]? (End)
%C Answer: Yes, of course: The set of injections satisfying [1] and [2] is not empty, so there's a lexicographically least element. More concretely, it starts with the same 23 terms a(0..22) which are known to be minimal, but after a(22) = 41 it has to go on with a(23) = 41 + 23 = 64, since choosing "-" here necessarily yields a non-injective sequence. See A171884. - _M. F. Hasler_, Apr 01 2019
%C 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
%C 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
%C Of all the sequences in the OEIS, this one is my favorite to listen to. Click the "listen" button at the top, set the instrument to "103. FX 7 (Echoes)", click "Save", and open the MIDI file with a program like QuickTime Player 7. - _N. J. A. Sloane_, Aug 08 2017
%C This sequence cycles clockwise around the OEIS logo. - _Ryan Brooks_, May 09 2020
%D Alex Bellos and Edmund Harriss, Visions of the Universe (2016), Unnumbered pages. Includes Harriss's illustration of the first 65 steps drawn as a spiral.
%D Benjamin Chaffin, N. J. A. Sloane, and Allan Wilks, On sequences of Recaman type, paper in preparation, 2006.
%D Bernardo Recamán Santos, letter to N. J. A. Sloane, Jan 29 1991
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H N. J. A. Sloane, <a href="/A005132/b005132.txt">The first 100000 terms</a>
%H Alex Bellos and Brady Haran, <a href="https://www.youtube.com/watch?v=FGC5TdIiT9U">The Slightly Spooky Recamán Sequence</a>, Numberphile video, 2018.
%H Alex Bellos and Brady Haran, <a href="/A005132/a005132_1.png">Edmund Harriss's illustration of first 62 steps drawn as a spiral</a>, snapshot from Numberphile video "The Slightly Spooky Recamán Sequence" (2018) at 2:37 minutes. [Included with permission of the authors] See also the Harriss link below.
%H Benjamin Chaffin, <a href="/A005132/a005132.png">Log-log plot of first 10^230 terms</a>
%H Benjamin Chaffin, <a href="/A005132/a005132_1.txt">Status Report</a>, Jan 25 2018.
%H Fabio Deelan Cunden, Marilena Ligabò, and Tommaso Monni, <a href="https://arxiv.org/abs/2301.13555">Random matrices associated to Young diagrams</a>, arXiv:2301.13555 [math.PR], 2023. See p. 7.
%H Michael De Vlieger, video of first 1200 steps of the <a href="https://youtu.be/xl1yLPWypmo">Recamán sequence</a>, with audio accompaniment generated by aspects of the sequence. Oct 12, 2019.
%H GBnums, <a href="https://www.echolalie.org/echolaliste/gbnums/eois.htm">A nice OEIS sequence</a>
%H Gordon Hamilton, <a href="http://www.youtube.com/watch?v=mQdNaofLqVc">Wrecker Ball Sequences</a>, Video, 2013
%H Edmund Harriss, <a href="/A005132/a005132_1.pdf">The first 65 steps drawn as a spiral</a>
%H Nick Hobson, <a href="/A005132/a005132.py.txt">Python program for this sequence</a>
%H Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625, Dec 08, 2020
%H C. L. Mallows, <a href="/A005132/a005132.jpg">Plot (jpeg) of first 10000 terms</a>
%H C. L. Mallows, <a href="/A005132/a005132.ps">Plot (postscript) of first 10000 terms</a>
%H Joseph Samuel Myers, Richard Schroeppel, Scott R. Shannon, N. J. A. Sloane, and Paul Zimmermann, <a href="http://arxiv.org/abs/2004.14000">Three Cousins of Recaman's Sequence</a>, arXiv:2004:14000 [math.NT], April 2020.
%H Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Recaman%27s_sequence.svg">First 172 items in a coordinate system</a> [This is a graph of the start of A005132 rotated clockwise by 90 degs. - _N. J. A. Sloane_, Mar 23 2012]
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polcrt01.jpg">Illustration of initial terms of A001057, A005132, A000217</a>, 2012.
%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polrec01.jpg">Illustration of initial terms</a>, 2012.
%H Omar E. Pol, <a href="https://oeis.org/plot2a?name1=A005132&name2=A000004&tform1=untransformed&tform2=untransformed&shift=0&radiop1=matp&drawlines=true">Lateral view of a 3D-model whose front view is formed by spirals</a>, 2022 (using Plot 2 A005132 vs A000004)
%H Bernardo Recamán Santos and N. J. A. Sloane, <a href="/A005132/a005132.pdf">Correspondence</a>, 1991.
%H Scott R. Shannon, <a href="/A005132/a005132_2.png">Illustration of the first 2000 terms plotted as steps on a 2D square (Ulam) spiral</a>. The colors are graduated across the spectrum from red to violet to show the relative step order.
%H Muhammad Khubab Siddique, <a href="https://www.researchgate.net/publication/344659758_Mathematics-II_B_Ed_Programme">Sequence and Series-I</a>, Unit 8, Mathematics-II, Dept. of Sci. Ed., Allama Iqbal Open Univ. (Islamabad, Pakistan, 2020), 281-313.
%H N. J. A. Sloane, <a href="http://neilsloane.com/doc/sg.txt">My favorite integer sequences</a>, in Sequences and their Applications (Proceedings of SETA '98).
%H N. J. A. Sloane, <a href="/A005132/a005132.txt">FORTRAN program for A005132, A057167, A064227, A064228</a>
%H N. J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, pp. 10, 12-13.
%H Alex van den Brandhof, <a href="https://pyth.eu/uploads/user/ArchiefPDF/Pyth55-2.pdf">Een bizarre rij</a>, Pythagoras, 55ste Jaargang, Nummer 2, Nov 2015 (Article in Dutch about this sequence, see page 19 and back cover).
%H <a href="/index/Rea#Recaman">Index entries for sequences related to Recamán's sequence</a>
%F a(k) = A000217(k) - 2*Sum_{i=1..n} A057166(i), for A057166(n) <= k < A057166(n+1). - _Christopher Hohl_, Jan 27 2019
%e 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.
%p 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
%p A005132 := proc(n)
%p option remember; local a, found, j;
%p if n = 0 then return 0 fi;
%p a := procname(n-1) - n ;
%p if a <= 0 then return a+2*n fi;
%p found := false;
%p for j from 0 to n-1 while not found do
%p found := procname(j) = a;
%p od;
%p if found then a+2*n else a fi;
%p end:
%p seq(A005132(n), n=0..70); # _R. J. Mathar_, Apr 01 2012 (reformatted by _Peter Luschny_, Jan 06 2019)
%t 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
%t (* Second program: *)
%t 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 *)
%o (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_
%o (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
%o (MATLAB)
%o function a=A005132(m)
%o % m=max number of terms in a(n). Offset n:0
%o a=zeros(1,m);
%o for n=2:m
%o B=a(n-1)-(n-1);
%o C=0.^( abs(B+1) + (B+1) );
%o D=ismember(B,a(1:n-1));
%o a(n)=a(n-1)+ (n-1) * (-1)^(C + D -1);
%o end
%o % _Adriano Caroli_, Dec 26 2010
%o (Haskell)
%o import Data.Set (Set, singleton, notMember, insert)
%o a005132 n = a005132_list !! n
%o a005132_list = 0 : recaman (singleton 0) 1 0 where
%o recaman :: Set Integer -> Integer -> Integer -> [Integer]
%o recaman s n x = if x > n && (x - n) `notMember` s
%o then (x-n) : recaman (insert (x-n) s) (n+1) (x-n)
%o else (x+n) : recaman (insert (x+n) s) (n+1) (x+n)
%o -- _Reinhard Zumkeller_, Mar 14 2011
%o (Python)
%o l=[0]
%o for n in range(1, 101):
%o x=l[n - 1] - n
%o if x>0 and not x in l: l+=[x, ]
%o else: l+=[l[n - 1] + n]
%o print(l) # _Indranil Ghosh_, Jun 01 2017
%o (Python)
%o def recaman(n):
%o seq = []
%o for i in range(n):
%o if(i == 0): x = 0
%o else: x = seq[i-1]-i
%o if(x>=0 and x not in seq): seq+=[x]
%o else: seq+=[seq[i-1]+i]
%o return seq
%o print(recaman(1000)) # _Ely Golden_, Jun 14 2018
%o (Python)
%o from itertools import count, islice
%o def A005132_gen(): # generator of terms
%o a, aset = 0, set()
%o for n in count(1):
%o yield a
%o aset.add(a)
%o a = b if (b:=a-n)>=0 and b not in aset else a+n
%o A005132_list = list(islice(A005132_gen(),30)) # _Chai Wah Wu_, Sep 15 2022
%Y 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).
%Y A065056 gives partial sums, A160356 gives first differences.
%Y A row of A066201.
%Y Cf. A171884 (injective variant).
%Y See A324784, A324785, A324786 for the "low points".
%Y Cf. A330791, A331659, A331670.
%K nonn,nice,hear,look
%O 0,3
%A _N. J. A. Sloane_ and _Simon Plouffe_, May 16 1991
%E _Allan Wilks_, Nov 06 2001, computed 10^15 terms of this sequence. At this point all the numbers below 852655 had appeared, but 852655 = 5*31*5501 was missing.
%E After 10^25 terms of A005132 the smallest missing number is still 852655. - _Benjamin Chaffin_, Jun 13 2006
%E Even after 7.78*10^37 terms, the smallest missing number is still 852655. - _Benjamin Chaffin_, Mar 28 2008
%E Even after 4.28*10^73 terms, the smallest missing number is still 852655. - _Benjamin Chaffin_, Mar 22 2010
%E Even after 10^230 terms, the smallest missing number is still 852655. - _Benjamin Chaffin_, 2018
%E Changed "positive" in definition to "nonnegative". - _N. J. A. Sloane_, May 04 2020
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