cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A005132 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.

Original entry on oeis.org

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, 17, 43, 16, 44, 15, 45, 14, 46, 79, 113, 78, 114, 77, 39, 78, 38, 79, 37, 80, 36, 81, 35, 82, 34, 83, 33, 84, 32, 85, 31, 86, 30, 87, 29, 88, 28, 89, 27, 90, 26, 91, 157, 224, 156, 225, 155
Offset: 0

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Author

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

Keywords

Comments

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
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
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)
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
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
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
This sequence cycles clockwise around the OEIS logo. - Ryan Brooks, May 09 2020

Examples

			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.

References

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

Links

Crossrefs

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.
Cf. A171884 (injective variant).
See A324784, A324785, A324786 for the "low points".
Cf. A330791, A331659, A331670.

Programs

  • 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
  • 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
  • Maple
    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, found, j;
    if n = 0 then return 0 fi;
    a := procname(n-1) - n ;
    if a <= 0 then return a+2*n fi;
    found := false;
    for j from 0 to n-1 while not found do
        found := procname(j) = a;
    od;
    if found then a+2*n else a fi;
end:
seq(A005132(n), n=0..70); # R. J. Mathar, Apr 01 2012 (reformatted by Peter Luschny, Jan 06 2019)
  • Mathematica
    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
(* Second program: *)
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 *)
RecamanSeq[i_Integer] := Fold[With[{lst=Last@#, len=Length@#}, Append[#, If[lst > len && !MemberQ[#, lst - len], lst - len, lst + len]]] &, {0}, Range@i]; RecamanSeq[10^5] (* Mikk Heidemaa, Nov 02 2024 *)
  • 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<M. F. Hasler, May 11 2008, updated M. F. Hasler, Nov 03 2014
  • Python
    l=[0]
for n in range(1, 101):
    x=l[n - 1] - n
    if x>0 and not x in l: l+=[x, ]
    else: l+=[l[n - 1] + n]
print(l) # Indranil Ghosh, Jun 01 2017
  • Python
    def recaman(n):
  seq = []
  for i in range(n):
    if(i == 0): x = 0
    else: x = seq[i-1]-i
    if(x>=0 and x not in seq): seq+=[x]
    else: seq+=[seq[i-1]+i]
  return seq
print(recaman(1000)) # Ely Golden, Jun 14 2018
  • Python
    from itertools import count, islice
def A005132_gen(): # generator of terms
    a, aset = 0, set()
    for n in count(1):
        yield a
        aset.add(a)
        a = b if (b:=a-n)>=0 and b not in aset else a+n
A005132_list = list(islice(A005132_gen(),30)) # Chai Wah Wu, Sep 15 2022

Formula

a(k) = A000217(k) - 2*Sum_{i=1..n} A057166(i), for A057166(n) <= k < A057166(n+1). - Christopher Hohl, Jan 27 2019

Extensions

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.
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, 2018
Changed "positive" in definition to "nonnegative". - N. J. A. Sloane, May 04 2020

A331659 Fixed points in the Recamán sequence; k such that A005132(k) = k.

Original entry on oeis.org

0, 1, 1520, 9317, 31221, 325374626148, 535755688021, 1404720439053, 3883018238329, 16166305650060
Offset: 1

Views

Author

Jud McCranie, Jan 23 2020

Keywords

Comments

This is the intersection of A064568 and A330791.
No more terms < 6.46*10^13. - James Ewens, Sep 27 2024

Examples

			A005132(1520) = 1520, so 1520 is in the sequence.

Links

Crossrefs

Cf. A005132, A064568, A330791.

Extensions

a(7)-a(10) from James Ewens, Sep 27 2024

A336830 The Sydney Opera House sequence: a(0) = 0, a(1) = 1; for n > 0, a(n) = min(a(n-1)/n if n|a(n-1), a(n-1)-n) where a(n) is nonnegative and not already in the sequence. Otherwise a(n) = min(a(n-1)+n, a(n-1)*n) where a(n) is not already in the sequence. Otherwise a(n) = a(n-1) + n.

Original entry on oeis.org

0, 1, 2, 5, 9, 4, 10, 3, 11, 20, 30, 19, 7, 91, 77, 62, 46, 29, 47, 28, 8, 168, 146, 123, 99, 74, 48, 21, 49, 78, 108, 139, 107, 140, 106, 71, 35, 72, 34, 73, 33, 1353, 1311, 1268, 1224, 1179, 1133, 1086, 1038, 989, 939, 888, 836, 783, 729, 674, 618, 561, 503, 444, 384, 323, 261
Offset: 0

Views

Author

Scott R. Shannon, Aug 05 2020

Keywords

Comments

This sequence is similar to the Recamán sequence A005132 except that division and multiplication by n are also permitted. This leads to larger variations in the values of the terms while minimizing the repetition of previously visited terms.
To determine a(n), initially a(n-1)-n is calculated if a(n-1)-n is nonnegative, along with a(n-1)/n if n|a(n-1). If one or both of these have not already appeared in the sequence then a(n) is set to the minimum of these candidates. If neither are candidates then both a(n-1)+n and a(n-1)*n are calculated. If one or both of these have not already appeared in the sequence then a(n) is set to the minimum of these candidates. If neither are candidates, i.e., all of a(n-1)-n, a(n-1)/n, a(n-1)+n, a(n-1)*n are either invalid or have already been visited, then a(n) = a(n-1)+n. However for the first 100 million terms no instance is found where all four options are unavailable, although it is unknown if this eventually occurs for very large n.
For the first 100 million terms the smallest value not appearing is 6. As with the Recamán sequence it is unknown if this and other small unseen terms eventually appear. The largest term is a(50757703) = 6725080695952885. In the same range, division, subtraction, addition, and multiplication are chosen for the next term 38, 99965692, 34188, and 81 times, respectively.

Examples

			a(2) = 2. As a(1) = 1, which is not divisible by 2 nor greater than 2, a(2) must be the minimum of 1*2=2 and 1+2=3, so multiplication is chosen.
a(5) = 4. As a(4) = 9, which is not divisible by 5, and 4 has not appeared previously in the sequence, a(5) = a(4)-5 = 9-5 = 4.
a(82) = 52. As a(81) = 4264 one candidate is 4264-82 = 4182. However 82|4264 and 4264/82 = 52. Neither of these candidates has previously appeared in the sequence, but 52 is the minimum of the two. This is the first time a division operation is used for a(n).

Links

Crossrefs

Cf. A005132, A113880, A171884, A330791, A362398, A362399.

Programs

  • Python
    global arr
arr = []
def a(n):
    # Case 1
    if n == 0:
        return 0
    a_prev = arr[-1]
    cand = []
    # Case 2
    x = a_prev - n
    y = a_prev / n
    if x > 0 and not x in arr:
        cand.append(x)
    if y == int(y) and not y in arr:
        cand.append(y)
    if cand != []:
        return min(cand)
    # Case 3
    cand = []
    x = a_prev + n
    y = a_prev * n
    if not x in arr:
        cand.append(x)
    if not y in arr:
        cand.append(y)
    if cand != []:
        return min(cand)
    # Case 4
    return a_prev + n
def seq(n):
    for i in range(n):
        print("{}, ".format(a(i)), end="")
        arr.append(a(i))
seq(60)
# Christoph B. Kassir, Apr 08 2022
  • Python
    from itertools import count, islice
def A336830(): # generator of terms
    aset, an, oo = {0, 1}, 1, float('inf')
    yield from [0, 1]
    for n in count(2):
        v1, v2 = an - n if an >= n else oo, an//n if an%n == 0 else oo
        v = min((vi for vi in [v1, v2] if vi not in aset), default=oo)
        if v != oo: an = v
        else:
            v3, v4 = an+n, an*n
            v = min((vi for vi in [v3, v4] if vi not in aset), default=oo)
            if v != oo: an = v
            else: an = an+n
        yield an
        aset.add(an)
print(list(islice(A336830(), 60))) # Michael S. Branicky, Apr 15 2023
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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