A335923 -id:A335923 - cp's OEIS frontend

A335924 A variation on Recamán's sequence (A005132): a(0) = 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence; otherwise, a(n) = a(n-1) + ceiling(n/2) if not already in the sequence and a(n) = a(n-1) + n if already in the sequence.

0, 1, 2, 4, 6, 9, 3, 7, 11, 16, 21, 10, 22, 29, 15, 23, 31, 14, 32, 13, 33, 12, 34, 46, 58, 71, 45, 18, 46, 17, 47, 63, 79, 96, 62, 27, 63, 26, 64, 25, 65, 24, 66, 88, 44, 67, 90, 43, 91, 42, 92, 41, 93, 40, 94, 39, 95, 38, 96, 37, 97, 36, 98, 35, 99, 132, 165

Offset: 0

Ya-Ping Lu, Jun 29 2020

nonn

In this sequence, a forward step of ceiling(n/2) is added if a(n) - n is negative and a(n-1) + ceiling(n/2) is not already in the sequence. As a result, both the number of distinct numbers and the number of distinct numbers as a percentage of the biggest number in the sequence (called "coverage") are increased.
The smallest missing numbers, h1, from the first m terms of the sequence, given as h1(m), are: 3(6), 5(11097), 57(49518), 149(92113), 159(124908), ... All integers less than or equal to h1 can be found in the first m+1 terms of the sequence.
The number of consecutive numbers (Nc, from 0 to Nc-1), the distinct numbers (Nd), the biggest number (a_max), and the "coverage" for n=0~1000000 in the sequences with different forward and backward steps are given below:
Sequence  Backward  Forward      Nc     Nd       a_max     coverage
A005132   -n        +n           1355   736749   5946126   12.39%
A335923   -n        +n/2          620   694811   4350902   15.97%
"B"       -n        +n/4          577   696167   3132344   22.23%
A335924   -n        +n/2, +n      160   813204   5698099   14.27%
"C"       -n        +n/4, +n/2   1330   779087   3757167   20.74%
"D"       -2n, -n   +n/2, +n       24   901949   3639015   24.79%
"E"       -2n, -n   +n/4, +n/2   3414   817174   3128675   26.12%

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Altug Alkan, Andrew R. Booker, and Florian Luca, On a recursively defined sequence involving the prime counting function, arXiv:2006.08013 [math.NT], 2020.

Cf. A005132, A335923.

Nest[Append[#1, If[And[#3 >= 0, FreeQ[#1, #3]], #3, If[FreeQ[#1, #4], #4, #1[[-1]] + #2]]] & @@ {#1, #2, #1[[-1]] - #2, #1[[-1]] + Ceiling[#2/2]} & @@ {#, Length@ #} &, {0}, 10^5] (* Michael De Vlieger, Sep 09 2020 *)

import math
n_max = 1000000
a_last = 0
list1 = [a_last]
print(0)
for n in range(1, n_max+1):
    m1 = a_last - n
    m2 = a_last + math.ceil(n/2)
    if m1 >= 0 and m1 not in list1:
        a = m1
    elif m2 not in list1:
        a = m2
    else:
        a = a_last + n
    list1.append(a)
    print(a)
    a_last = a

from itertools import count, islice
def A335924_gen(): # generator of terms
    a, aset = 0, set()
    for n in count(1):
        yield a
        aset.add(a)
        a = b if (b:=a-n)>=0 and b not in aset else c if (c:=(n+1>>1)+a) not in aset else a+n
A335924_list = list(islice(A335924_gen(),70)) # Chai Wah Wu, Sep 15 2022

cp's OEIS Frontend

A335924 A variation on Recamán's sequence (A005132): a(0) = 0, a(n) = a(n-1) - n if nonnegative and not already in the sequence; otherwise, a(n) = a(n-1) + ceiling(n/2) if not already in the sequence and a(n) = a(n-1) + n if already in the sequence.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Python

Python