A057167 -id:A057167 - cp's OEIS frontend

A064294 Term at which last number of height n occurs in Recamán's sequence A005132.

1, 4, 5, 16, 31, 64, 131, 222, 403, 770, 1409, 2652, 4825, 9078, 16773, 30768, 56827, 99734, 181653, 328002, 588583, 1032996, 1787013, 3220128, 5771203, 10201340, 18394609, 32144188, 58055311, 101762018, 173367175, 302844912, 511518279, 904032692

Offset: 1

N. J. A. Sloane, Sep 26 2001

nonn

The height of a term in A005132 = number of addition steps - number of subtraction steps to produce it.

Cf. A005132, A064290, A064292, A064293. This sequence and A064492 keep pretty close together.

A057166 Indices of subtraction steps in Recamán's sequence A005132.

Original entry on oeis.org

4, 8, 10, 12, 14, 16, 20, 22, 23, 25, 27, 29, 31, 35, 37, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 68, 70, 72, 74, 76, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 103, 105, 107, 109, 110, 111, 116, 118, 119, 120, 122, 124, 126, 128

Offset: 1

N. J. A. Sloane, Sep 14 2000

nonn

Nick Hobson, Python program for this sequence
Colin Mallows, Plot (jpeg) of first 10000 terms of A005132
Colin Mallows, Plot (postscript) of first 10000 terms of A005132
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057165, A057167.

A245340 Smallest m such that A125717(m) = n, or -1 if n never appears.

Original entry on oeis.org

0, 1, 4, 2, 8, 21, 3, 5, 18, 16, 14, 12, 10, 6, 1518, 32, 58, 30, 184, 28, 7, 26, 9, 11, 13, 15, 17, 19, 102, 51, 100, 49, 98, 47, 96, 45, 94, 43, 92, 41, 90, 39, 88, 37, 86, 35, 84, 20, 24, 22, 505, 81, 2510, 79, 166, 77, 296, 75, 501, 73, 162, 71, 498, 69

Offset: 0

Reinhard Zumkeller, Jul 21 2014

nonn

Conjecture: a(n) is never -1.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Ferenc Adorjan, Some characteristics of Leroy Quet's permutation sequences
N. J. A. Sloane, Log-log plot of A370956 vs A370959 (shows terms in A125717 that take the longest to appear).

Cf. A245394, A245395, A057167.

For RECORDS see A370956 and A370959.

import Data.IntMap (singleton, member, (!), insert)
a245340 n = a245340_list !! n
a245340_list = 0 : f [1..] [1..] 0 (singleton 0 0) where
   f us'@(u:us) vs'@(v:vs) w m
     | u `member` m = (m ! u) : f us vs' w m
     | otherwise    = g (reverse[w-v,w-2*v..1] ++ [w+v,w+2*v..]) where
     g (x:xs) = if x `member` m then g xs else f us' vs x $ insert x v m

from itertools import count
def A245340(n):
    a, aset = 0, set()
    for m in count(1):
        if a==n: return m-1
        aset.add(a)
        a = next(a for a in count(a%m,m) if a not in aset) # Chai Wah Wu, Mar 13 2024

A078758 Term in A064389 (variation (4) of Recamán's sequence) where n appears, or 0 if n never appears.

Original entry on oeis.org

1, 4, 2, 35, 33, 3, 5, 16, 14, 12, 10, 8, 6, 188, 58, 186, 184, 23, 25, 7, 9, 11, 13, 15, 17, 182, 48, 46, 50, 1774, 1770, 1772, 1786, 1784, 1766, 40, 1764, 32, 34, 36, 22, 20, 18, 24, 26, 204, 202, 200, 198, 196, 194, 192, 190, 1758, 1762, 1760, 79, 77, 1782, 602

Offset: 1

Reiner Martin, Jan 08 2003

nonn

Is it ever 0?
If all a(n)>0, then this is the inverse permutation of A064389. - M. F. Hasler, Nov 03 2014
After 10^12 terms of A064389, the first unknown term of this sequence is 5191516. - Benjamin Chaffin, Oct 09 2016

Benjamin Chaffin, Table of n, a(n) for n = 1..20000 [First 6655 terms from Nick Hobson]
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

Cf. A064389, A005132, A057167.

for(i=1,9e9,for(j=1,#a064389,a064389[j]==i&&print1(j",")+next(2));break) \\ The terms of A064389 must be stored in the vector a064389. - M. F. Hasler, Nov 03 2014

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).

Original entry on oeis.org

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))

A335300 a(n) is the least k such that A335299(k) = n, or -1 if no such k exists.

Original entry on oeis.org

0, 1, 4, 2, 71, 73, 3, 5, 16, 14, 12, 10, 8, 6, 31, 29, 27, 25, 23, 438, 7, 9, 11, 13, 15, 17, 59, 57, 520, 410, 56, 54, 52, 50, 48, 46, 44, 42, 40, 38, 692, 22, 20, 18, 28, 30, 32, 97, 424, 234, 96, 94, 92, 90, 88, 86, 240, 4518, 296, 294, 464, 462, 19, 21

Offset: 0

Rémy Sigrist, May 31 2020

nonn

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Logarithmic scatterplot of the sequence for n = 0..1000000
Rémy Sigrist, C++ program for A335300
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057167, A335299.

A118202 Inverse of A118201, or -1 if n does not occur in A118201.

Original entry on oeis.org

0, 1, 4, 2, 1715, 31, 3, 5, 16, 14, 12, 10, 8, 6, 476, 101, 99, 97, 23, 303, 7, 9, 11, 13, 15, 17, 52, 50, 48, 46, 44, 42, 40, 38, 36, 34, 32, 30, 28, 683, 681, 22, 20, 18, 24, 77, 699, 697, 703, 695, 693, 691, 689, 687, 1723, 1725, 1721, 267, 269, 2686, 67, 263, 19, 21

Offset: 0

Franklin T. Adams-Watters, Apr 14 2006

nonn

Conjecture that a(n) is never -1.

Cf. A118201, A057167.

A293273 a(n) is the smallest positive k <> n such that f(k) is divisible by f(n) where f = A005132, or 0 if no such k exists.

Original entry on oeis.org

2, 3, 8, 3, 9, 35, 43, 15, 20, 11, 28, 7, 32, 21, 83, 15, 69, 26, 152, 24, 116, 47, 44, 20, 48, 18, 43, 59, 30, 63, 20, 104, 41, 71, 39, 75, 72, 35, 35, 36, 33, 79, 92, 83, 96, 87, 100, 91, 245, 95, 239, 67, 276, 19, 119, 63, 109, 57, 103, 51, 185, 45, 139, 35, 145, 86, 415, 84, 192, 82, 184, 80, 180, 78, 176

Offset: 1

Altug Alkan, Oct 10 2017

Conjecture: a(n) > 0 for all n.

			a(6) = 35 because A005132(35) = 78 is divisible by A005132(6) = 13 and 78 is the smallest positive number which is not equal to 6 with this property.

Robert Israel, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Logarithmic scatterplot of the first 100000 terms
Rémy Sigrist, Scatterplot of the first 100000 terms of the first difference

Cf. A005132, A057167.

N:= 10^4: # to use A005132(n) for n = 1..N
S:= {0}:
A5132:= Array(0..N):
A5132[0]:= 0:
for n from 1 to N do
  v:= A5132[n-1]-n;
  if v < 0 or member(v,S) then v:= A5132[n-1]+n fi;
  A5132[n]:= v;
  S:= S union {v};
od:
f:= proc(n) local k;
  for k from 1 to N do
    if k <> n and A5132[k] mod A5132[n] = 0 then return k fi
  od:
0
end proc:
Res:= NULL:
for n from 1 do
  v:= f(n);
  if v = 0 then break fi;
  Res:= Res,v;
od:
Res; # Robert Israel, Oct 10 2017

A072349 a(n) = m such that A064388(m) = n, or 0 if no such m exists.

Original entry on oeis.org

1, 4, 2, 934, 932, 3, 5, 16, 14, 12, 10, 8, 6, 31, 29, 27, 25, 23, 236, 7, 9, 11, 13, 15, 17, 64, 62, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 40

Offset: 1

David Wasserman, Jul 17 2002

nonn

Next term, a(39), is either 0 or greater than 400000. My guess is that this sequence is not computable, i.e., there is some n that does not occur in A064388, but it is impossible to prove this.

Index entries for sequences related to Recamán's sequence

Cf. A057167.

A355646 Term in Recamán's sequence A005132 where n appears for the last time, or -1 if n never appears.

Original entry on oeis.org

0, 1, 4, 2, 131, 129, 3, 5, 16, 14, 12, 10, 8, 6, 31, 29, 27, 25, 23, 99734, 7, 9, 11, 13, 15, 17, 64, 62, 60, 58, 56, 54, 52, 50, 48, 46, 44, 42, 40, 38, 111, 22, 24, 26, 28, 30, 32, 222, 220, 218, 216, 214, 212, 210, 208, 206, 204, 202, 200, 198, 196, 181653

Offset: 0

Iain Fox, Jul 11 2022

nonn

Same sequence as A057167 until a(42) = 24.
Let k be the largest member of A057167 before A057167(n). Trivially, if A057167(n) > k, a(n) = A057167(n).
If every nonnegative integer appears in A005132, then -1 does not appear in this sequence.

Cf. A005132, A057167.

cp's OEIS Frontend

A064294 Term at which last number of height n occurs in Recamán's sequence A005132.

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A057166 Indices of subtraction steps in Recamán's sequence A005132.

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A245340 Smallest m such that A125717(m) = n, or -1 if n never appears.

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A078758 Term in A064389 (variation (4) of Recamán's sequence) where n appears, or 0 if n never appears.

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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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A335300 a(n) is the least k such that A335299(k) = n, or -1 if no such k exists.

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A118202 Inverse of A118201, or -1 if n does not occur in A118201.

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A293273 a(n) is the smallest positive k <> n such that f(k) is divisible by f(n) where f = A005132, or 0 if no such k exists.

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A072349 a(n) = m such that A064388(m) = n, or 0 if no such m exists.

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A355646 Term in Recamán's sequence A005132 where n appears for the last time, or -1 if n never appears.

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