A187921 -id:A187921 - cp's OEIS frontend

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

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

A187922 Positions k of addition steps in Recamán's sequence where A005132(k-1)-k = A005132(m) for some 0 < m < k.

Original entry on oeis.org

6, 7, 9, 18, 19, 21, 33, 34, 36, 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, 287, 288, 290, 292, 294, 296, 298, 300, 302, 304, 339, 340, 342, 344, 346, 348, 350

Offset: 1

Reinhard Zumkeller, Mar 17 2011

nonn

Subsequence of A057165; A005132(a(n)-1) - a(n) = A005132(A187943(n));
A005132(a(n)) = A005132(a(n)-1) + a(n);
See A187921 for the other positions of addition steps in A005132.

			a(5) = 19: A005132(19-1) = 43 and 43-19>0, but the term 24=43-19 is already in A005132, therefore A005132(19)=43+19=62; A187943(5)=15 and A005132(15)=24.

Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, C++ program for A187943
Index entries for sequences related to Recamán's sequence

Cf. A005132, A057165, A187921, A187943, A333548, A333552.

import Data.Set (Set, singleton, member, insert)
a187922 n = a187922_list !! (n-1)
a187922_list = r (singleton 0) 1 0 where
   r :: Set Integer -> Integer -> Integer -> [Integer]
   r s n x | x <= n           = r (insert (x+n) s) (n+1) (x+n)
           | (x-n) `member` s = n : r (insert (x+n) s) (n+1) (x+n)
           | otherwise        = r (insert (x-n) s) (n+1) (x-n)
(C++) See Links section.

Added condition "0 < m" to definition. See A333552. - N. J. A. Sloane, May 04 2020

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

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

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A187922 Positions k of addition steps in Recamán's sequence where A005132(k-1)-k = A005132(m) for some 0 < m < k.

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

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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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