A064289 -id:A064289 - cp's OEIS frontend

A078759 Height of n-th term in A064389 (variation (4) of Recamán's sequence).

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, 7, 8, 7, 8, 7, 6, 5, 6, 5, 6, 7, 8, 7, 6, 7, 8, 9, 8, 7, 6, 7, 6, 7, 6, 7, 8, 9, 8, 9, 8, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 9, 8, 7, 8, 9, 10, 9, 10, 9, 8, 7, 8, 7, 8, 9, 10, 9, 10, 9, 10, 9, 8, 9, 10, 11, 10, 11, 10, 9, 8, 9, 8, 9, 10

Offset: 1

Reiner Martin, Jan 08 2003

nonn

Nick Hobson, Table of n, a(n) for n=1..10000
Nick Hobson, Python program for this sequence

Cf. A005132, A064289, A064389.

The height of a term in A064389 = number of addition steps - number of subtraction steps to produce it.

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.

A276439 Partial sums of A276438.

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, 7, 8, 7, 6, 8, 9, 8, 9, 8, 7, 9, 10, 9, 8, 7, 8, 7, 9, 10, 9, 8, 10, 11, 10, 9, 8, 7, 8, 7, 8, 7, 8, 7, 8, 9, 10, 9, 10, 9, 8, 10, 11, 10, 9, 8, 9, 8, 9, 10, 11, 10, 9, 8, 9, 8, 10, 11, 10, 11, 10, 9, 8, 9, 10, 11, 10, 11, 12, 13, 12, 11, 10, 9, 8, 9, 10

Offset: 0

Antti Karttunen, Sep 04 2016

nonn

Antti Karttunen, Table of n, a(n) for n = 0..10000

Cf. A274647, A276438
Cf. also A064289, A078759.
Differs from A078759 for the first time at n=24, where a(24)=7 while A078759(24)=6.

a(0) = 0; for n >= 1, a(n) = A276438(n) + a(n-1).

A269831 Least term of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 2, 6, 8, 14, 26, 4, 47, 92, 111, 181, 150, 371, 361, 781, 828, 366, 19

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.

Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293, A269830, A269832.

A269832 Greatest term of height n in Recamán's sequence A005132.

Original entry on oeis.org

1, 3, 7, 13, 25, 46, 91, 164, 286, 515, 962, 1744, 3137, 5810, 10319, 18953, 35079, 63237

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.

Index entries for sequences related to Recamán's sequence

Cf. A064289, A064290, A064293, A269830, A269831.

A325933 Partial sums of A325931.

Original entry on oeis.org

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

Offset: 0

Allan C. Wechsler, Sep 08 2019

This sequence is the "tier profile" of A076042, much as A064289 provides a similar profile of Recamán's Sequence (A005132). In each case, the base sequence jumps around between adjacent "tiers", each of which changes value relatively slowly (compared to the difference between entries in different tiers). As in Recamán's sequence, each tier rises for a while, then crests, and finally falls close to zero, at which point that tier becomes extinct. The tier structure is salient when viewing the graph of such a sequence.

This sequence is less chaotic than A064289, since in A076042 only two tiers are active at time, and the sequence alternates between the two until the lower tier stops, at which point a double increase inaugurates a new tier.

			a(20) = a(19) + A325931(20) = 5 + 1 = 6.

Alois P. Heinz, Table of n, a(n) for n = 0..65536

Cf. A076042, A325931.

b:= proc(n) option remember; `if`(n=0, 0, (t->
      t+`if`(tAlois P. Heinz, Sep 08 2019

b[n_] := b[n] = If[n == 0, 0, b[n - 1] + If[b[n - 1] < n^2, n^2, -n^2]];
b /@ Range[0, 105] // Differences // Sign // Accumulate // Prepend[#, 0]& (* Jean-François Alcover, Nov 30 2020 *)

a(0) = 0; a(n) = a(n-1) + A325931(n).

A383730 a(0) = 0, a(n) = a(n-1) + A002260(n) * (-1)^(n-1) if not already in the sequence, otherwise a(n) = a(n-1) - A002260(n) * (-1)^(n-1).

Original entry on oeis.org

0, 1, 2, 4, 3, 5, 8, 9, 7, 10, 6, 5, 7, 4, 8, 13, 12, 14, 11, 15, 20, 26, 25, 27, 24, 28, 23, 29, 22, 21, 19, 16, 20, 15, 21, 14, 22, 21, 23, 20, 24, 19, 25, 32, 40, 49, 48, 50, 47, 51, 46, 52, 45, 53, 44, 54, 55, 57, 60, 64, 59, 65, 58, 66, 75, 85, 74, 73, 71

Offset: 0

Markel Zubia, May 06 2025

This sequence is a bidirectional form of Recamán's sequence.
Another way to define the sequence: starting at 0, take steps of size 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, ... alternating left and right while avoiding repeated values (negative values are allowed).
The sequence is unbounded either above or below.
Conjecture: the sequence is unbounded both above and below.
Conjecture: each integer appears finitely often.
It exhibits a mix of chaotic and periodic behavior, including long plateaus and sudden large jumps.
Around term 25000, the sequence settles near -3000 in a visually fractal structure. After ~1.85 million terms, it appears to settle again near -500000. Astonishingly, after ~66.7 million steps, it jumps sharply from around -500000 to +4.51 million.

			a(1) = 0 + 1.
a(2) = 0 + 1 + 1 = 2, since 0 + 1 - 1 = 0 already appears in the sequence, as a(0) = 0.
a(6) = 0 + 1 + 1 + 2 - 1 + 2 + 3 = 8.

Markel Zubia, Table of n, a(n) for n = 0..9999
Markel Zubia, Plot of the first 100k terms
Markel Zubia, Close-up of the fractal-like pattern
Markel Zubia, Plot of the first 10M terms
Markel Zubia, Plot of the first 1B terms

Cf. A005132.
Other bidirectional extensions of Recamán's sequence: A063733, A079053, A064288, A064289, A064387, A064388, A064389, A228474.
Cf. A002260.

def a(n):
    t, k, curr = 1, 1, 0
    seen = set()
    for i in range(n):
        seen.add(curr)
        step = (-1)**i * t
        if curr + step not in seen:
            curr = curr + step
        else:
            curr = curr - step
        t += 1
        if t > k:
            t, k = 1, k + 1
    return curr

cp's OEIS Frontend

A078759 Height of n-th term in A064389 (variation (4) of Recamán's sequence).

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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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A276439 Partial sums of A276438.

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A269831 Least term of height n in Recamán's sequence A005132.

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A269832 Greatest term of height n in Recamán's sequence A005132.

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A325933 Partial sums of A325931.

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A383730 a(0) = 0, a(n) = a(n-1) + A002260(n) * (-1)^(n-1) if not already in the sequence, otherwise a(n) = a(n-1) - A002260(n) * (-1)^(n-1).

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Python