A005132 -id:A005132 - cp's OEIS frontend

A261573 A variation of Recamán's sequence A005132: Define a(0) = 0, and for n > 0, a(n) = a(n-1) - (n+2) if positive and not already in the sequence, otherwise a(n) = a(n-1) + (n+2).

0, 3, 7, 2, 8, 1, 9, 18, 28, 17, 5, 18, 4, 19, 35, 52, 34, 15, 35, 14, 36, 13, 37, 12, 38, 11, 39, 10, 40, 71, 103, 70, 104, 69, 33, 70, 32, 71, 31, 72, 30, 73, 29, 74, 120, 167, 119, 168, 118, 67, 119, 66, 120, 65, 121, 64, 6, 65, 125, 186, 124, 61, 125, 60, 126, 59, 127, 58, 128, 57

Offset: 0

Freddy Barrera, Aug 24 2015

nonn

As in Recamán's sequence, terms are repeated, the first being 18 = a(7) = a(11).
More generally, for k >= 0, a_k(0) = 0, and for n > 0, a_k(n) = a_k(n-1) - (n+k) if positive and not already in the sequence, otherwise a_k(n) = a_k(n-1) + (n+k).
For k = 0, this is Recamán's sequence A005132.

Freddy Barrera, Table of n, a(n) for n = 0..10000
Freddy Barrera, Line plot of the first 1000 terms.
Freddy Barrera, Scatter plot of the first 1000 terms.
Freddy Barrera, Line plot of the first 1000 terms when k = 0, 1, 2, ..., 9
Freddy Barrera, Scatter plot of the first 1000 terms when k = 0, 1, 2, ..., 9

Cf. A005132, A057167.

f[s_List] := Block[{a = s[[-1]], len = Length@ s}, Append[s, If[a > len + 1 && ! MemberQ[s, a - len - 2], a - len - 2, a + len + 2]]]; Nest[f, {0}, 70] (* Robert G. Wilson v, Sep 08 2015 *)

def sequence(n, k):
    """For n > 0 and k >= 0, generates the first n terms of the sequence"""
    A, a = {0}, 0
    yield a
    for n in range(1, n + 1):
        a = a - (n + k)
        if a > 0 and a not in A:
            A.add(a)
            yield a
        else:
            a = a + 2 * (n + k)
            A.add(a)
            yield a
# List of the first 1000 terms of the sequence with k = 2.
list(sequence(1000, 2))

A269830 Number of terms of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 2, 6, 11, 22, 34, 61, 115, 220, 397, 681, 1329, 2430, 4561, 8116, 14848, 24878

Offset: 1

Danny Rorabaugh, Mar 05 2016

The height (A064289) of a term in Recamán's sequence (A005132) = number of addition steps - number of subtraction steps to produce it.

Danny Rorabaugh, All terms of height n<=18 in Recamán's sequence
Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293.

A274648 A variation on Recamán's sequence (A005132): a(n) is the first positive number of the form a(n-1)-nk, k>0 not already in the sequence; and if no such number exists, then a(n) is the first number of the form a(n-1)+nk, k>0 not already in the sequence.

Original entry on oeis.org

0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24, 8, 25, 43, 5, 45, 66, 44, 67, 19, 69, 17, 71, 15, 73, 103, 72, 40, 106, 38, 108, 36, 110, 34, 112, 32, 114, 30, 116, 28, 118, 26, 120, 168, 119, 169, 16, 68, 121, 175, 65, 177, 63, 179, 61, 181, 59, 183

Offset: 0

Max Barrentine, Aug 12 2016

nonn

Is this a permutation of the natural numbers?
The inverse is: 0, 1, 4, 2, 164, 19, 3, 5, 16, 14, 12, 10, 8, 6, 8228, 28, 51, 26, 158, 24, 7, 9, 11, 13, 15, 17, 46, 90, ..., . Robert G. Wilson v, Sep 07 2016
After 3.2*10^11 terms, the smallest number which has not appeared is 154. - Benjamin Chaffin, Oct 05 2016

Robert G. Wilson v, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Recamán's sequence

Cf. A273148 (inverse), A005132, A274647 (another variant).

f[s_List] := Block[{k = 1, l = s[[-1]], n = Length@ s}, While[ MemberQ[s, l - k*n] && l > k*n, k++]; If[l > k*n, Append[s, l - k*n], k = 1; While[ MemberQ[s, l + k*n], k++]; Append[s, l + k*n]]]; Nest[f, {0}, 60] (* Robert G. Wilson v, Sep 07 2016 *)

A330791 Values of k such that A005132(k) (the k-th number in the Recamán sequence) divides k.

Original entry on oeis.org

1, 4, 16, 58, 624, 1520, 8346, 9317, 31221, 240048, 297624, 778144, 2757880, 4670769, 7654224, 24206384, 54806653, 76521364, 260393208, 846807152, 19488644316, 58671741024, 105968380189, 140689013614, 325374626148, 535755688021, 1404720439053, 3116427665348, 3883018238329, 16166305650060

Offset: 1

Jud McCranie, Jan 23 2020

nonn

The intersection of A064568 and this sequence are the values of k such that A005132(k)=k.

No more terms < 6.46*10^13. - James Ewens, Sep 28 2024

			A005132(4)=2 and 2 divides 4, so 4 is in the sequence.

Index entries for sequences related to Recamán's sequence

Cf. A005132, A330788, A330789, A064568.

a(26)-a(30) from James Ewens, Sep 28 2024

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

Jud McCranie, Jan 23 2020

This is the intersection of A064568 and A330791.

No more terms < 6.46*10^13. - James Ewens, Sep 27 2024

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

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064568, A330791.

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

A334951 a(n) is the smallest candidate for the n-th term of Recamán's sequence A005132.

Original entry on oeis.org

0, -1, -1, 0, 2, -3, 1, 6, 12, 3, 11, 0, 10, -3, 9, -6, 8, -9, 7, 24, 42, 21, 41, 18, -6, 17, -9, 16, -12, 15, -15, 14, -18, 13, 45, 78, 42, 77, 39, 0, 38, -3, 37, -6, 36, -9, 35, -12, 34, -15, 33, -18, 32, -21, 31, -24, 30, -27, 29, -30, 28, -33, 27, -36, 26, -39, 25, 90, 156, 87, 155, 84

Offset: 0

Omar E. Pol, May 17 2020

sign

For n > 0 and after A005132(n-1) the algorithm of Recamán's sequence first explores if a(n) is a valid number to be its next term. If a(n) is nonnegative and not already in the sequence A005132 then a(n) is accepted, so A005132(n) = a(n), otherwise a(n) is rejected and A005132(n) = A005132(n-1) + n, not a(n).

For an illustration of initial terms see the diagram in A334950.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to Recamán's sequence

Bisection of A334950.

Cf. A005132, A334952.

a(0) = 0; for n > 0, a(n) = A005132(n-1) - n.

A064284 Number of times n appears in Recamán's sequence A005132.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 0

N. J. A. Sloane, Sep 24 2001

nonn

Index entries for sequences related to Recamán's sequence

Cf. A005132.

a(0)=1 inserted by Sean A. Irvine, Jun 26 2023

A064569 Quotients A005132(k)/k for k=A064568(n).

Original entry on oeis.org

1, 2, 2, 3, 2, 2, 2, 2, 1, 2, 1, 4, 1, 2, 5, 3, 2, 3, 4, 3, 4, 3, 1, 1, 1, 1, 4, 1, 3, 5

Offset: 1

N. J. A. Sloane, Oct 16 2001

No more terms < 6.46*10^13. - James Ewens, Sep 27 2024

			From _Seiichi Manyama_, May 02 2020: (Start)
Let b(n) = A005132(A064568(n)).
   n | A064568(n) |    b(n) | a(n)
-----+------------+---------+------
   1 |          1 |       1 |    1
   2 |          3 |       6 |    2
   3 |         11 |      22 |    2
   4 |         21 |      63 |    3
   5 |         39 |      78 |    2
   6 |         76 |     152 |    2
   7 |        248 |     496 |    2
   8 |        844 |    1688 |    2
   9 |       1520 |    1520 |    1
  10 |       2752 |    5504 |    2
  11 |       9317 |    9317 |    1
  12 |      17223 |   68892 |    4
  13 |      31221 |   31221 |    1
  14 |      57071 |  114142 |    2
  15 |      99741 |  498705 |    5
  16 |     589932 | 1769796 |    3 (End)

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064568, A331659, A333548.

a(1)=1 inserted by Seiichi Manyama, May 02 2020

a(22)-a(30) added by James Ewens, Sep 27 2024

A065052 Let R(n) = n-th term of Recamán's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of r.

Original entry on oeis.org

0, 1, 0, 2, 2, 1, 6, 4, 3, 1, 0, 10, 10, 9, 9, 8, 8, 7, 5, 2, 0, 19, 18, 18, 17, 17, 16, 16, 15, 15, 14, 14, 13, 11, 8, 6, 3, 1, 0, 38, 38, 37, 37, 36, 36, 35, 35, 34, 34, 33, 33, 32, 32, 31, 31, 30, 30, 29, 29, 28, 28, 27, 27, 26, 26, 25, 23, 20, 18, 15, 13, 10, 8, 5, 3

Offset: 1

Allan Wilks, Nov 06 2001

nonn

Index entries for sequences related to Recamán's sequence

Cf. A065051, A005132.

A123483 Second order Recamán's sequence: a(0) = 0; for n > 0, a(n) = a(n-1) - A005132(n) if that number is positive and not already in the sequence, otherwise a(n-1) + A005132(n).

Original entry on oeis.org

0, 1, 4, 10, 8, 15, 2, 22, 34, 13, 24, 46, 36, 59, 50, 26, 18, 43, 86, 148, 106, 169, 128, 110, 68, 51, 94, 78, 122, 107, 62, 48, 94, 173, 60, 138, 252, 175, 136, 58, 20, 99, 136, 56, 92, 11, 46, 128, 162, 79, 112, 28, 60, 145, 114, 200, 170, 83, 54, 142, 170, 81, 108, 198

Offset: 0

Franklin T. Adams-Watters, Sep 28 2006

nonn

I conjecture that this sequence contains every natural number. (Even though through n=10000, we still haven't seen 3; we are still occasionally seeing small numbers.) This sequence has an interesting graph.

The smallest n such that a(n) = 3, 6, 12 are, respectively, 4729925, 5808155, 2093396. The following numbers less than 100 do not appear in the sequence for n <= 10^7: 7, 17, 19, 21, 23, 25, 27, 29, 35, 39, 41, 44, 45, 57, 61, 65, 67, 70, 71, 73, 77, 87, 91, 95. - Nick Hobson, Feb 18 2007

Franklin T. Adams-Watters, The first 10000 terms
Nick Hobson, Python program for this sequence

Cf. A005132.

cp's OEIS Frontend

A261573 A variation of Recamán's sequence A005132: Define a(0) = 0, and for n > 0, a(n) = a(n-1) - (n+2) if positive and not already in the sequence, otherwise a(n) = a(n-1) + (n+2).

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A269830 Number of terms of height n in Recamán's sequence A005132.

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A274648 A variation on Recamán's sequence (A005132): a(n) is the first positive number of the form a(n-1)-n*k, k>0 not already in the sequence; and if no such number exists, then a(n) is the first number of the form a(n-1)+n*k, k>0 not already in the sequence.

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A330791 Values of k such that A005132(k) (the k-th number in the Recamán sequence) divides k.

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

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A334951 a(n) is the smallest candidate for the n-th term of Recamán's sequence A005132.

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Formula

A064284 Number of times n appears in Recamán's sequence A005132.

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A064569 Quotients A005132(k)/k for k=A064568(n).

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A065052 Let R(n) = n-th term of Recamán's sequence A005132; write R(n) = q*n + r with 0 <= r < n; sequence gives values of r.

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A123483 Second order Recamán's sequence: a(0) = 0; for n > 0, a(n) = a(n-1) - A005132(n) if that number is positive and not already in the sequence, otherwise a(n-1) + A005132(n).

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A274648 A variation on Recamán's sequence (A005132): a(n) is the first positive number of the form a(n-1)-nk, k>0 not already in the sequence; and if no such number exists, then a(n) is the first number of the form a(n-1)+nk, k>0 not already in the sequence.