A160357 -id:A160357 - cp's OEIS frontend

A119632 Lengths of successive runs in A160357, where a run here means a string of alternating terms.

1, 1, 3, 1, 11, 1, 4, 10, 1, 4, 28, 1, 10, 24, 1, 8, 1, 2, 1, 1, 4, 1, 9, 4, 1, 2, 36, 1, 12, 4, 1, 2, 1, 3, 28, 1, 10, 52, 1, 18, 1, 32, 1, 12, 15, 38, 1, 14, 32, 1, 12, 1, 44, 1, 16, 1, 148, 1, 50, 7, 22, 1, 8, 3, 4, 1, 2, 70, 1, 24, 1, 114, 1, 42, 1, 200, 1, 68, 6, 1, 2, 13

Offset: 1

N. J. A. Sloane and Allan Wilks, Jun 10 2006

nonn

Gives a highly compressed version of A005132.
The encoding of Recamán's sequence a(n) = A005132 using A119632 is easy - A119632 counts runs of alternating i(n)'s, where i(n) = (a(n)-a(n-1))/n = A160357(n).
Note that i(n) is always +1 or -1.  Each run ends when i(n) = i(n+1).
Here is pseudo-code to reconstruct Recamán's sequence from A119632, which we will call I(n):
        a(0) = 0
        n = 1
        i = 1
        for k = 1..oo {
                for j = 1..I(k) {
                        a(n) = a(n-1) + n*i
                        n = n+1
                        i = -i
                }
                i = -i
        }
The gzipped file attached to A119632 represents the first 1470117206801829 terms of A005132. The longest run of alternating i(n)'s (maximal value found so far in A119632) is 232144588914. There are 64094657 runs encoded in the gzipped file.

			A160357 begins 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; ..., where semicolons demark the successive runs.

Allan Wilks, Table of n, a(n) for n = 1..100000
Allan Wilks, The first 64094657 terms (gzipped). (A large file. This encodes the first 1470117206801829 terms of A005132!)
Index entries for sequences related to Recamán's sequence

Cf. A005132, A160357.

Entry expanded by N. J. A. Sloane, Jul 15 2011

A064289 Height of n-th term in Recamán's sequence A005132.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 4, 5, 6, 7, 6, 7, 6, 5, 6, 5, 6, 5, 6, 5, 6, 5, 6, 7, 8, 7, 8, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 9, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7, 8, 7

Offset: 0

N. J. A. Sloane, Sep 25 2001

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

Partial sums of A160357. - Allan C. Wechsler, Sep 08 2019

			A005132 begins 1, 3, 6, 2, 7, 13, 20, 12, ... and these terms have heights 1, 2, 3, 2, 3, 4, 5, 4, ...

Alois P. Heinz, Table of n, a(n) for n = 0..100000
Nick Hobson, Python program for this sequence
N. J. A. Sloane, FORTRAN program for A005132, A057167, A064227, A064228
Index entries for sequences related to Recamán's sequence

Cf. A005132, A064288, A064290, A064292, A064293, A064294, A160357.

g:= proc(n) is(n=0) end:
b:= proc(n) option remember; local t;
      if n=0 then 0 else t:= b(n-1)-n; if t<=0 or g(t)
      then t:= b(n-1)+n fi; g(t):= true; t fi
    end:
a:= proc(n) option remember; `if`(n=0, 0,
       a(n-1)+signum(b(n)-b(n-1)))
    end:
seq(a(n), n=0..120);  # Alois P. Heinz, Sep 08 2019

g[n_] := n == 0;
b[n_] := b[n] = Module[{t}, If[n == 0, 0, t = b[n - 1] - n; If[t <= 0 || g[t], t = b[n - 1] + n]; g[t] = True; t]];
a[n_] := a[n] = If[n == 0, 0, a[n - 1] + Sign[b[n] - b[n - 1]]];
a /@ Range[0, 100] (* Jean-François Alcover, Apr 11 2020, after Alois P. Heinz *)

a(0)=0 prepended by Allan C. Wechsler, Sep 08 2019

A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 5, 3, 4, 4, 5, 11, 13, 7, 8, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17

Offset: 1

Omar E. Pol, Mar 24 2012

Consider a toothpick structure formed by L-toothpicks connected by their endpoints. The endpoints of the L-toothpicks are placed on the main diagonal of the first quadrant. At stage 1 we place an L-toothpick with one of its endpoints on the origin. At stage n we place an L-toothpick of size n. The L-toothpicks are placed alternately, on one or another sector of the first quadrant, trying to make the structure have an exposed endpoint closest to the origin. The total length of all L-toothpicks after the n-th stage is A002378(n). The value of x and y of the endpoint of the structure after the n-th stage is equal to the n-th term of Recamán's sequence A005132(n). Note that we can get other illustrations of initial terms of Recamán's sequence by replacing each L-toothpick by a Q-toothpick or by a semicircumference. This structure is also one of the three views of the three-dimensional model for Recamán's sequence. For more information about L-toothpicks and Q-toothpicks, see A172310 and A187210.

			The summands are the size of the L-toothpicks:
a(1) = 1.
a(2) = 1 + 2 = 3.
a(3) = 2 + 3 = 5.
a(4) = 3.
a(5) = 4.
a(6) = 4.
a(7) = 5.
a(8) = 5 + 6 = 11.
a(9) = 6 + 7 = 13.
a(10) = 7.

First differences of A210607.

Cf. A005132, A064289, A119632, A139250, A160356, A160357, A171175, A171178, A172310, A187210, A210604, A210605, A210608-A210613.

A160356 First differences of Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 3, -4, 5, 6, 7, -8, 9, -10, 11, -12, 13, -14, 15, -16, 17, 18, 19, -20, 21, -22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, 33, 34, -35, 36, -37, -38, 39, -40, 41, -42, 43, -44, 45, -46, 47, -48, 49, -50, 51, -52, 53, -54, 55, -56, 57, -58, 59, -60, 61, -62, 63

Offset: 1

M. F. Hasler, Jun 03 2009

sign

Cf. A005132, A057165, A057166, A065056, A160357.

a(n) = A005132(n)-A005132(n-1) = n*A160357(n).

As a set, A160356 = A057165 union -A057166.

A076213 2*a(n)-1 = sign(A005132(n+1)-A005132(n)).

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0

Offset: 0

Benoit Cloitre, Nov 03 2002

nonn

Characteristic function of A057165 - 1. - M. F. Hasler, Jun 03 2009

Index entries for sequences related to Recamán's sequence

Conjecture: let s(n)=sum(k=1, n, a(k)), then lim n ->infinity s(n)/n = 1/2; for any n, 2*s(n) > n; let v(n)=2*s(n)-n, then v(n)/log(n) is bounded and sum(k=1, n, v(k)) is asymptotic to c*n*log(n) with 1 < c < 3/2.

a(n) = 1-A160351(n+1) = (A160357(n)+1)/2. - M. F. Hasler, Jun 03 2009

Added initial value a(0)=1. - M. F. Hasler, Jun 03 2009

A160351 Characteristic function of A057166.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 1

M. F. Hasler, Jun 03 2009

nonn

Index entries for sequences related to Recamán's sequence

a(n) = (1-A160357(n))/2 = 1-A076213(n-1).

A210608 Number of intersections after n-th stage in the structure mentioned in A210606 using semicircumferences and counting the superposed intersections several times.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 14, 22, 28, 36, 43

Offset: 1

Omar E. Pol, Mar 25 2012

The structure mentioned in A210606 is a model for the visualization of Recamán's sequence A005132.

First differs from A210609 at a(24).

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064289, A119632, A160356, A160357, A171175, A171178, A210606, A210607, A210609-A210613.

A210609 Number of intersections after n-th stage in the structure mentioned in A210606 using semicircumferences and counting the superposed intersections only once.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 14, 21, 27

Offset: 1

Omar E. Pol, Mar 25 2012

The structure mentioned in A210606 is a model for the visualization of Recamán's sequence A005132.

First differs from A210608 at a(24).

Index entries for sequences related to Recamán's sequence

Cf. A005132, A064289, A119632, A160356, A160357, A171175, A171178, A210606, A210607, A210608, A210610-A210613.

A210610 Number of semicircumferences in the n-th spiral of the structure mentioned in A210606.

Original entry on oeis.org

3, 11, 4, 10, 4, 28, 10, 24, 8, 2, 4, 9, 4, 2, 36, 12, 4, 2, 3, 28, 10, 52, 18, 32, 12, 15, 38, 14, 32, 12, 44, 16, 148, 50, 7, 22, 8, 3, 4, 2, 70, 24, 114, 42, 200, 68, 6, 2, 13

Offset: 1

Omar E. Pol, Mar 25 2012

The structure mentioned in A210606 is a model for the visualization of Recamán's sequence A005132.

Index entries for sequences related to Recamán's sequence

Numbers > 1 in A119632.

Cf. A005132, A064289, A160356, A160357, A171175, A171178, A210606-A210609, A210611-A210613.

A366911 a(n) = (A364054(n+1) - A364054(n)) / prime(n) (where prime(n) denotes the n-th prime number).

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, -3, 2, -2, 1, -1, 1, -1, 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, -1, 1, -1, 1, -1, 1

Offset: 1

Rémy Sigrist, Oct 27 2023

sign

a(n) is the number of steps of size prime(n) in going from A364054(n) to A364054(n+1).

			a(7) = (A364054(8) - A364054(7)) / prime(7) = (19 - 2) / 17 = 1.

N. J. A. Sloane, Table of n, a(n) for n = 1..9999
Rémy Sigrist, PARI program

Cf. A160357, A364054, A366912 (partial sums).

nn = 2^16; c[] := False; m[] := 0; j = 1; c[0] = c[1] = True;
  Monitor[Do[p = Prime[n - 1]; r = Mod[j, p];
    While[Set[k, p m[p] + r ]; c[k], m[p]++];
    Set[{a[n - 1], c[k], j}, {(k - j)/p, True, k}], {n, 2, nn + 1}], n];
Array[a, nn] (* Michael De Vlieger, Oct 27 2023 *)

PARI
```
See Links section.
```

from itertools import count, islice
from sympy import nextprime
def A366911_gen(): # generator of terms
    a, aset, p = 1, {0,1}, 2
    while True:
        k, b = divmod(a,p)
        for i in count(-k):
            if b not in aset:
                aset.add(b)
                a, p = b, nextprime(p)
                yield i
                break
            b += p
A366911_list = list(islice(A366911_gen(),30)) # Chai Wah Wu, Oct 27 2023

cp's OEIS Frontend

A119632 Lengths of successive runs in A160357, where a run here means a string of alternating terms.

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A064289 Height of n-th term in Recamán's sequence A005132.

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Maple

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A210606 Length of the n-th edge of an L-toothpick structure which gives Recamán's sequence A005132.

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A160356 First differences of Recamán's sequence A005132.

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A076213 2*a(n)-1 = sign(A005132(n+1)-A005132(n)).

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A160351 Characteristic function of A057166.

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A210608 Number of intersections after n-th stage in the structure mentioned in A210606 using semicircumferences and counting the superposed intersections several times.

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A210609 Number of intersections after n-th stage in the structure mentioned in A210606 using semicircumferences and counting the superposed intersections only once.

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A210610 Number of semicircumferences in the n-th spiral of the structure mentioned in A210606.

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A366911 a(n) = (A364054(n+1) - A364054(n)) / prime(n) (where prime(n) denotes the n-th prime number).

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