A135287 -id:A135287 - cp's OEIS frontend

A090895 a(1)=1 then a(n)=a(n-1)/2 if a(n-1) is even, a(n)=a(n-1)+n otherwise.

1, 3, 6, 3, 8, 4, 2, 1, 10, 5, 16, 8, 4, 2, 1, 17, 34, 17, 36, 18, 9, 31, 54, 27, 52, 26, 13, 41, 70, 35, 66, 33, 66, 33, 68, 34, 17, 55, 94, 47, 88, 44, 22, 11, 56, 28, 14, 7, 56, 28, 14, 7, 60, 30, 15, 71, 128, 64, 32, 16, 8, 4, 2, 1, 66, 33, 100, 50, 25, 95, 166, 83, 156, 78, 39

Offset: 1

Benoit Cloitre, Feb 25 2004

nonn

Does a(n)=1 for infinitely many values of n ?
It seems that the answer is yes (see A185038). The number a(n) is always in the range on 1 to 3*a(n), and there is an average of 2 addition steps for every 5 steps. In order to reach '1', the sequence must reach a power of two after an addition step, which is likely to happen on an exponential basis. [Sergio Pimentel, Mar 01 2012]
a(A208852(n)) = n and a(m) != n for m < A208852(n); A185038(a(n)) = 1. [Reinhard Zumkeller, Mar 02 2012]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A135287, A185038.

a090895 n = a090895_list !! (n-1)
a090895_list = 1 : f 2 1 where
   f x y = z : f (x + 1) z where
        z = if m == 0 then y' else x + y; (y',m) = divMod y 2
-- Reinhard Zumkeller, Mar 02 2012

nxt[{n_,a_}]:={n+1,If[EvenQ[a],a/2,a+n+1]}; Transpose[NestList[nxt,{1,1},80]][[2]] (* Harvey P. Dale, Aug 25 2015 *)

a(n)=if(n<2,1,if(a(n-1)%2,a(n-1)+n,a(n-1)/2))

sum(k=1, n, a(k)) seems to be asymptotic to c*n^2 where c=0.57....

A135294 a(n) = 3*a(n-1)+n if a(n-1) is not divisible by 2, or a(n) = a(n-1)/2 otherwise.

Original entry on oeis.org

1, 4, 2, 1, 7, 26, 13, 46, 23, 78, 39, 128, 64, 32, 16, 8, 4, 2, 1, 22, 11, 54, 27, 104, 52, 26, 13, 66, 33, 128, 64, 32, 16, 8, 4, 2, 1, 40, 20, 10, 5, 56, 28, 14, 7, 66, 33, 146, 73, 268, 134, 67, 253, 812, 406, 203, 665, 2052, 1026, 513, 1599, 4858, 2429, 7350, 3675, 11090, 5545, 16702, 8351, 25122, 12561

Offset: 1

Ctibor O. Zizka, Dec 04 2007

nonn

a(n)=a(0)*(3^(n-i))/(2^i) + c where c is in the range (0..sum(i*3^(n-i))). Sum(i*3^(n-i)) for i=1 to n equals A001793 (coefficients of Chebyshev polynomials). Max a(n) = 3^n*(a(0)/3^i*2^i + 9/4) - ((2*n+5)/4) which for large n gives max a(n) ~ 2.25*3^n - n/2. - Ctibor O. Zizka, Dec 26 2007

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A135287.

nxt[{n_,a_}]:={n+1,If[OddQ[a],3a+n,a/2]}; NestList[nxt,{1,1},70][[;;,2]] (* Harvey P. Dale, Oct 01 2024 *)

Corrected and extended by Harvey P. Dale, Oct 01 2024

A345877 a(1) = 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = n - a(n-1).

Original entry on oeis.org

1, 1, 2, 1, 4, 2, 1, 7, 2, 1, 10, 5, 8, 4, 2, 1, 16, 8, 4, 2, 1, 21, 2, 1, 24, 12, 6, 3, 26, 13, 18, 9, 24, 12, 6, 3, 34, 17, 22, 11, 30, 15, 28, 14, 7, 39, 8, 4, 2, 1, 50, 25, 28, 14, 7, 49, 8, 4, 2, 1, 60, 30, 15, 49, 16, 8, 4, 2, 1, 69, 2, 1, 72, 36, 18, 9, 68, 34, 17, 63, 18, 9, 74, 37, 48, 24, 12, 6, 3

Offset: 1

Altug Alkan, Jun 28 2021

Let a_i(1) = 1 and a_i(n) = a_i(n-1)/(i+1) if a_i(n-1) is divisible by i+1, otherwise a_i(n) = n - a_i(n-1). This sequence is a_1(n) and A345886 is a_2(n).

Conjecture: a_i(n) hits every positive integers infinitely many times for all i >= 1.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A090895, A135287, A345886.

a[1] = 1; a[n_] := a[n] = If[EvenQ[a[n - 1]], a[n - 1]/2, n - a[n - 1]]; Array[a, 100] (* Amiram Eldar, Jun 29 2021 *)
nxt[{n_,a_}]:={n+1,If[EvenQ[a],a/2,n+1-a]}; NestList[nxt,{1,1},90][[;;,2]] (* Harvey P. Dale, Aug 31 2023 *)

q=vector(100); q[1]=1; for(n=2, #q, q[n] = if(q[n-1]%2, n-q[n-1], q[n-1]/2)); q

A134440 a(0)=1; for n > 0, a(n) = a(n-1) + prime(n) if a(n-1) is odd, else a(n) = a(n-1)/2.

Original entry on oeis.org

1, 3, 6, 3, 10, 5, 18, 9, 28, 14, 7, 38, 19, 60, 30, 15, 68, 34, 17, 84, 42, 21, 100, 50, 25, 122, 61, 164, 82, 41, 154, 77, 208, 104, 52, 26, 13, 170, 85, 252, 126, 63, 244, 122, 61, 258, 129, 340, 170, 85, 314, 157, 396, 198, 99, 356, 178, 89, 360, 180, 90, 45

Offset: 0

Ctibor O. Zizka, Jan 18 2008

nonn

LFSR with primes.

Is it true that Lim a(n)/prime(n) < square root(3)?

T. Herlestam, On functions of linear shift register sequences. Springer Lecture notes in computer sciences, ISBN 978-3-540-16468-5.

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Georg Schmidt and Vladimir R. Sidorenko, Linear Shift-Register Synthesis for Multiple Sequences of Varying Length, arXiv:cs/0605044 [cs.IT], 2001.
Boaz Tsaban and Uzi Vishne, Efficient linear feedback shift registers with maximal period, arXiv:cs/0304010 [cs.CR], 2003.

Cf. A000040, A135287.

A134440 := proc(n)
    option remember;
    if n =0 then
        1;
    elif type(procname(n-1),'odd') then
        procname(n-1)+ithprime(n) ;
    else
        procname(n-1)/2 ;
    end if;
end proc: # R. J. Mathar, Jun 20 2021

nxt[{n_,a_}]:={n+1,If[OddQ[a],a+Prime[n+1],a/2]}; Transpose[ NestList[ nxt,{0,1},70]][[2]] (* Harvey P. Dale, Jan 12 2016 *)

Offset corrected by R. J. Mathar, Jun 20 2021

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A090895 a(1)=1 then a(n)=a(n-1)/2 if a(n-1) is even, a(n)=a(n-1)+n otherwise.

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A135294 a(n) = 3*a(n-1)+n if a(n-1) is not divisible by 2, or a(n) = a(n-1)/2 otherwise.

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A345877 a(1) = 1, a(n) = a(n-1)/2 if a(n-1) is even, otherwise a(n) = n - a(n-1).

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A134440 a(0)=1; for n > 0, a(n) = a(n-1) + prime(n) if a(n-1) is odd, else a(n) = a(n-1)/2.

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