A187922 -id:A187922 - cp's OEIS frontend

A187943 A057167(A005132(A187922(n) - 1) - A187922(n)).

1, 3, 2, 5, 15, 9, 6, 30, 20, 17, 63, 57, 51, 45, 35, 21, 98, 92, 66, 72, 38, 110, 103, 109, 25, 31, 10, 16, 2, 126, 36, 169, 163, 157, 151, 145, 139, 45, 35, 101, 182, 21, 32, 133, 187, 105, 123, 193, 23, 29, 8, 14, 3, 2, 75, 284, 278, 272, 266, 260, 254, 248, 226, 232, 100, 260, 254, 248, 226, 232, 306, 138, 144, 150, 156, 162, 168, 34, 65, 117, 228, 234

Offset: 1

Reinhard Zumkeller, Mar 17 2011

Let x = A187922(n) and r = A005132(x-1) - x, by definition of A187922: r > 0 and r occurs earlier in Recamán's sequence: a(n) = A057167(r) or r = A005132(a(n)), hence A005132(x) <> r but A005132(x) = A005132(x-1) - x.

			n = 5: x = A187922(5) = 19, r = A005132(19-1) - 19 = 43 - 19 = 24 = A005132(15): a(5) = 15.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Recamán's sequence

Cf. A057165, A057167.

A057167 Term in Recamán's sequence A005132 where n appears for first 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, 20, 18, 28, 30, 32, 222, 220, 218, 216, 214, 212, 210, 208, 206, 204, 202, 200, 198, 196

Offset: 0

N. J. A. Sloane, Sep 14 2000

T. D. Noe, Table of n, a(n) for n=0..1642 (taking a(1355) from A064227)
Nick Hobson, Python program for this sequence
C. L. Mallows, Plot (jpeg) of first 10000 terms of A005132
C. L. Mallows, Plot (postscript) of first 10000 terms of A005132
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
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057165, A057166.

Cf. also A187943, A187922.

w := array(1..10000); for j from 1 to 100 do l := 0; for k from 1 to nops(a) do if a[k] = j then l := k; exit; fi; od: w[j] := l; od: s := [seq(w[j],j=1..100)]; # where a is an array formed from sequence A005132

A005132 = {0}; Do[If[(r = Last[A005132] - n) <= 0 || MemberQ[ A005132, r], r = r + 2n]; AppendTo[ A005132, r], {n, 1, 10^5}]; a[n_] := If[p = Position[ A005132, n]; p == {}, 0, p[[1, 1]] - 1]; Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Jul 18 2012 *)

first(n) = my(a=vector(n), r=[0]); while(#Set(a)Iain Fox, Jul 11 2022

from itertools import count, islice
def agen(): # generator of terms
    an, A005132set, inv, y = 0, {0}, {0: 0}, 0
    for n in count(1):
        t = an - n
        an = t if t >= 0 and t not in A005132set else an + n
        A005132set.add(an)
        inv[an] = n
        while y in inv: yield inv[y]; y += 1
print(list(islice(agen(), 61))) # Michael S. Branicky, Jul 12 2022

I conjecture a(n) is never -1 - but see A064227, A064228.

a(0)=0 added and escape clause value changed to -1 by N. J. A. Sloane, May 01 2020

A057165 Indices of addition steps in Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 7, 9, 11, 13, 15, 17, 18, 19, 21, 24, 26, 28, 30, 32, 33, 34, 36, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 66, 67, 69, 71, 73, 75, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 101, 102, 104, 106, 108, 112, 113, 114

Offset: 1

N. J. A. Sloane, Sep 14 2000

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10001
Nick Hobson, Python program for this sequence
C. L. Mallows, Plot (jpeg) of first 10000 terms of A005132
C. L. Mallows, Plot (postscript) of first 10000 terms of A005132
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057166, A057167.

Union of A187921 and A187922.

import Data.Set (Set, singleton, notMember, insert)
a057165 n = a057165_list !! n
a057165_list = r (singleton 0) 1 0 where
   r :: Set Integer -> Integer -> Integer -> [Integer]
   r s n x = if x > n && (x - n) `notMember` s
                then r (insert (x-n) s) (n+1) (x-n)
                else n : r (insert (x+n) s) (n+1) (x+n)
-- Reinhard Zumkeller, Mar 17 2011

Initial zero added by N. J. A. Sloane, May 01 2020

A187921 Positions k of addition steps in Recamán's sequence where A005132(k-1)<=k.

Original entry on oeis.org

1, 2, 3, 5, 11, 13, 15, 17, 24, 26, 28, 30, 32, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 112, 130, 132, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 248, 250

Offset: 1

Reinhard Zumkeller, Mar 17 2011

nonn

			a(10)=26: A005132(26-1) = 17 and 17-26<0, hence A005132(26) = 17+26 = 43.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Recamán's sequence

Subsequence of A057165;
A005132(a(n)-1) <= a(n); A005132(a(n)) = A005132(a(n)-1) + a(n);
see A187922 for the other positions of addition steps in A005132.

import Data.Set (Set, singleton, member, insert)
a187921 n = a187921_list !! (n-1)
a187921_list = r (singleton 0) 1 0 where
   r :: Set Integer -> Integer -> Integer -> [Integer]
   r s n x | x <= n           = n : r (insert (x+n) s) (n+1) (x+n)
           | (x-n) `member` s = r (insert (x+n) s) (n+1) (x+n)
           | otherwise        = r (insert (x-n) s) (n+1) (x-n)

A333548 Numbers k such that A005132(k-1) = k.

Original entry on oeis.org

3, 11, 39, 248, 844, 2752, 57071, 58056875

Offset: 1

N. J. A. Sloane, May 01 2020

Subtracting 1 from k gives the index of a term A005132(k-1) = k in Recamán's sequence A005132 such that subtracting k would reach 0. This is not permitted, so we must add k instead, obtaining A005132(k) = 2*k.

If A005132(k-1) = k, A005132(k) = 2*k. The converse is not always true. For example, A005132(75) = 228 and A005132(76) = 228 - 76 = 152. - Seiichi Manyama, May 02 2020

			A005132(10)=11, so 11 is a term (and A005132(11)=22).

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064568, A064569, A187921, A187922, A331659.

A333548_list, A005132_set, y = [], {0}, 0
for n in range(1,10**10):
    y -= n
    if y <= 0 or y in A005132_set:
        y += 2*n
    A005132_set.add(y)
    if y == n+1:
        A333548_list.append(y) # Chai Wah Wu, May 02 2020

a(8) from Chai Wah Wu, May 02 2020

A333552 List of numbers k defined by property that when the Recamán term A005132(k) is being computed, we are unable to subtract k from A005132(k-1) because, although A003132(k-1) >= k, the result of the subtraction, A005132(k-1)-k, is already in A005132.

Original entry on oeis.org

3, 6, 7, 9, 11, 18, 19, 21, 33, 34, 36, 39, 66, 67, 69, 71, 73, 75, 101, 102, 104, 106, 108, 113, 114, 115, 117, 121, 123, 125, 127, 133, 134, 172, 173, 175, 177, 179, 181, 183, 186, 188, 189, 190, 194, 224, 225, 227, 229, 231, 233, 236, 238, 240, 242, 244, 246, 248, 287, 288, 290, 292, 294, 296, 298, 300, 302, 304

Offset: 1

N. J. A. Sloane, May 03 2020

nonn

Positions k of addition steps in Recamán's sequence where A005132(k-1)-k = A005132(m) for some 0 <= m < k.

This is A187922 together with the terms in A333548. (The difference between A187922 and the present sequence is explained by the fact that originally A005132 began at 1 rather than 0.)

			After we have found A005132(6)=13, we attempt to subtract 7 from 13 to get a(7). However, this would give 6, which is a collision, since we already have A005132(3)=6. So 7 gets added to the current sequence.

N. J. A. Sloane, Table of n, a(n) for n = 1..22288
Index entries for sequences related to Recamán's sequence

Cf. A005132, A187921, A187922, A333548, A333549, A334494.

A334494 Positions k of addition steps in Recamán's sequence A005132 where A005132(k-1)

Original entry on oeis.org

1, 2, 5, 13, 15, 17, 24, 26, 28, 30, 32, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 112, 130, 132, 137, 139, 141, 143, 145, 147, 149, 151, 153, 155, 157, 159, 161, 163, 165, 167, 169, 171, 197, 199, 201, 203, 205, 207, 209, 211, 213, 215, 217, 219, 221, 223, 250

Offset: 1

N. J. A. Sloane, May 04 2020

nonn

This is A187921 with the terms in A333548 removed. (The difference between A187921 and the present sequence is explained by the fact that originally A005132 began at 1 rather than 0.)

Cf. A005132, A187921, A187922, A333548, A333552.

cp's OEIS Frontend

A187943 A057167(A005132(A187922(n) - 1) - A187922(n)).

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A057167 Term in Recamán's sequence A005132 where n appears for first time, or -1 if n never appears.

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A057165 Indices of addition steps in Recamán's sequence A005132.

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A187921 Positions k of addition steps in Recamán's sequence where A005132(k-1)<=k.

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A333548 Numbers k such that A005132(k-1) = k.

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Python

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A333552 List of numbers k defined by property that when the Recamán term A005132(k) is being computed, we are unable to subtract k from A005132(k-1) because, although A003132(k-1) >= k, the result of the subtraction, A005132(k-1)-k, is already in A005132.

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A334494 Positions k of addition steps in Recamán's sequence A005132 where A005132(k-1)

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