A081848 -id:A081848 - cp's OEIS frontend

A073941 a(n) = ceiling((Sum_{k=1..n-1} a(k)) / 2) for n >= 2 starting with a(1) = 1.

1, 1, 1, 2, 3, 4, 6, 9, 14, 21, 31, 47, 70, 105, 158, 237, 355, 533, 799, 1199, 1798, 2697, 4046, 6069, 9103, 13655, 20482, 30723, 46085, 69127, 103691, 155536, 233304, 349956, 524934, 787401, 1181102, 1771653, 2657479, 3986219, 5979328, 8968992

Offset: 1

Reinhard Zumkeller, Nov 20 2002

a(n) is the number of even integers that have n-1 digits when written in base 3/2. For example, there are 2 even integers that use three digits in base 3/2: 6 and 8: they are written as 210 and 212, respectively. - Tanya Khovanova and PRIMES STEP Senior group, Jun 03 2018
From Petros Hadjicostas, Jul 20 2020: (Start)
We describe Schuh's counting-off game (pp. 373-375 and 377-379). Assume m people are standing on a circle and they are labeled 1 through m (say clockwise). We start with the person labeled 1 and every 3rd person drops out (in a variation of the famous Josephus problem). The process is repeated until only one person is left.
This sequence describes those numbers m for which either the person labeled 1 or the person labeled 2 is the last survivor.
From a(4) = 2 to a(53) = 775795914 (see T. D. Noe's b-file), the values agree with those in Schuh (1968, p. 374) and Burde (1987, p. 207). a(54) = 1163693871 while both Schuh and Burde have 1063693871 (a difference in the 2nd digit starting on the left). a(55) = 1745540806 while both Schuch and Burde have 1595540806.
Schuh (1968) obtains the numbers in the following way. Suppose we know a(n) and the corresponding number i(n) of the last survivor (i(n) = 1 or 2). We multiply a(n) by 3/2 (cf. Burde's use of fractional bases).
If the product is an integer, that is a(n+1) and the corresponding last survivor is the same.
If the product is not an integer, then a(n+1) = floor(a(n)*3/2) if the last survivor i(n) = 2 (and the new last survivor is i(n+1) = 1), and a(n+1) = ceiling(a(n)*3/2) if the last survivor is i(n) = 1 (and the new last survivor is i(n+1) = 2).
Note that a(53) = 775795914 and a(54) = (3/2)*a(53) = 1163693871 (not 1063693871), so it seems Schuh did a mistake and Burde copied it. Also (3/2)*1163693871 = 1745540806.5. Since a(53) = 775795914 corresponds to number 2, we round down, i.e., a(54) = 1745540806 (and move to number 1). If, however, we multiply the incorrect 1063693871 by 3/2 and round down, we get Schuh and Burde's incorrect value 1595540806 for a(54).
Numbers a(n) that correspond to last survivors being number 1 are tabulated in A081614 while numbers a(n) that correspond to last survivors being number 2 are tabulated in A081615. (End)
a(n) is the number of times (n-1) appears in A061420. - Chinmaya Dash, Aug 19 2020

Fred Schuh, The Master Book of Mathematical Recreations, Dover, New York, 1968. [See Table 18, p. 374. Only the terms from a(6) = 4 forward are shown in the table. The table is definitely related to this sequence.]

T. D. Noe, Table of n, a(n) for n = 1..500
K. Burde, Das Problem der Abzählreime und Zahlentwicklungen mit gebrochenen Basen [The problem of counting rhymes and number expansions with fractional bases], J. Number Theory 26(2) (1987), 192-209. [The author deals with the representation of n in fractional bases k/(k-1) and its relation to counting-off games. Here k = 3. See the table on p. 207. See also the review in MathSciNet (MR0889384) by R. G. Stoneham.]
B. Chen, R. Chen, J. Guo, S. Lee et al, On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
Tom Edgar, Hailey Olafson, and James Van Alstine, Approximating the Fibonacci Sequence, Integers 16 (2016), #A63.

Same as log_2(A082125(n)), for n > 2. - Ralf Stephan, Apr 16 2002
Apart from initial term, same as A005428, which has further information.
a(n+4) = A079719(n)+2. Cf. A082416.
Partial sums for various start indices are in A006999, A061419, A061418. - Ralf Stephan, Apr 17 2003
Is this the same as A081848/3?
The constant c is (2/9)*K(3) (see A083286). - Ralf Stephan, May 29 2003
Cf. A081614, A081615.

a073941 n = a073941_list !! (n-1)
a073941_list = 1 : f [1] where
   f xs = x' : f (x':xs) where x' = (1 + sum xs) `div` 2
-- Reinhard Zumkeller, Oct 26 2011

f[s_] := Append[s, Ceiling[Plus @@ s/2]]; Nest[f, {1}, 41] (* Robert G. Wilson v, Jul 07 2006 *)

v=vector(100);s=v[1]=1;for(i=2,#v,s+=(v[i]=(s+1)\2));v \\ Charles R Greathouse IV, Feb 11 2011

from itertools import islice
def A073941_gen(): # generator of terms
    a, c = 1, 0
    yield 1
    while True:
        yield (a:=(c:=c+a)+1>>1)
A073941_list = list(islice(A073941_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n) = ceiling(c*(3/2)^n-1/2) where c = 0.3605045561966149591015446628665... - Benoit Cloitre, Nov 22 2002
If 2^m divides a(i) then 2^(m-1)*3^1 divides a(i+1) and so on... until finally, 3^m divides a(i+m). - Ralf Stephan, Apr 20 2003
a(n) = A081848(n)/3. - Tom Edgar, Jul 21 2014
a(n) = A005428(n-2). - Tanya Khovanova and PRIMES STEP Senior group, Jun 03 2018

A024629 n written in fractional base 3/2.

Original entry on oeis.org

0, 1, 2, 20, 21, 22, 210, 211, 212, 2100, 2101, 2102, 2120, 2121, 2122, 21010, 21011, 21012, 21200, 21201, 21202, 21220, 21221, 21222, 210110, 210111, 210112, 212000, 212001, 212002, 212020, 212021, 212022, 212210, 212211, 212212, 2101100, 2101101

Offset: 0

David W. Wilson

A246435(n) = (number of digits in a(n)) = A055642(a(n)). - Reinhard Zumkeller, Sep 05 2014
The number of positive even n such that a(n) has k+1 digits is A005428(k). - Glen Whitney, Jul 09 2017
The position of the rightmost "2" digit in a(3k), k >= 1, appears to be A087088(k). - Peter Munn, Jun 24 2020 [updated Peter Munn, Jul 14 2020 for new A087088 offset]

			Representations of the first few numbers are:
   0 =         0
   1 =         1
   2 =         2
   3 =       2 0
   4 =       2 1
   5 =       2 2
   6 =     2 1 0
   7 =     2 1 1
   8 =     2 1 2
   9 =   2 1 0 0
  10 =   2 1 0 1
  11 =   2 1 0 2
  12 =   2 1 2 0
  13 =   2 1 2 1
  14 =   2 1 2 2
  15 = 2 1 0 1 0
[extended and reformatted by _Peter Munn_, Jun 27 2020]

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Matvey Borodin, Hannah Han, Kaylee Ji, Tanya Khovanova, Alexander Peng, David Sun, Isabel Tu, Jason Yang, William Yang, Kevin Zhang, and Kevin Zhao, Variants of Base 3 over 2, arXiv:1901.09818 [math.NT], 2019.
B. Chen, R. Chen, J. Guo, S. Lee et al, On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
Michel Dekking, The Thue-Morse sequence in base 3/2, arXiv:2301.13563 [math.CO], 2023. See also J. Int. Seq., Vol. 26 (2023), Article 23.2.3.
Tanya Khovanova and Kevin Wu, Base 3/2 and Greedily Partitioned Sequences, arXiv:2007.09705 [math.NT], 2020.
J. S. Tanton, A collection of research problems. [archived version]

Cf. A081848, A087088, A246435 (string lengths), A244040 (digit sums).

a024629 0 = 0
a024629 n = 10 * a024629 (2 * n') + t where (n', t) = divMod n 3
-- Reinhard Zumkeller, Sep 05 2014

a:= proc(n) `if`(n<1, 0, irem(n, 3, 'q')+a(2*q)*10) end:
seq(a(n), n=0..45);  # Alois P. Heinz, Jun 19 2018

a[ n_] := If[ n < 1, 0, a[ Quotient[n, 3] 2] 10 + Mod[ n, 3]]; (* Michael Somos, Jun 18 2014 *)

{a(n) = if( n<1, 0, a(n\3 * 2) * 10 + n%3)}; /* Michael Somos, Jun 18 2014 */

def basepqExpansion(p,q,n):
    L, i = [n], 1
    while L[i-1] >= p:
        x=L[i-1]
        L[i-1]=x.mod(p)
        L.append(q*(x//p))
        i+=1
    L.reverse()
    return Integer(''.join(str(x) for x in L))
[basepqExpansion(3,2,n) for n in [0..40]] # Tom Edgar, Hailey R. Olafson, and James Van Alstine, Jun 17 2014; modified and corrected by G. C. Greubel, Aug 20 2019

To represent a number in base b, if a digit is >= b, subtract b and carry 1. In fractional base a/b, subtract a and carry b.

a(0)=0, a(3n+r) = 10*a(2n)+r for n >= 0 and r = 0, 1, 2. - Jianing Song, Oct 15 2022

Tanton link corrected by Charles R Greathouse IV, Oct 20 2008

A246435 Length of representation of n in fractional base 3/2.

Original entry on oeis.org

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

Offset: 0

Reinhard Zumkeller, Sep 05 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
Index entries for sequences related to fractional bases.

Cf. A024629, A055642, A070989, A081604, A081848 (run lengths), A244040.

a246435 n = if n < 3 then 1 else a246435 (2 * div n 3) + 1
-- Reinhard Zumkeller, Sep 05 2014

a[n_] := If[n < 3, 1, a[2 Quotient[n, 3]] + 1]; Array[a, 100, 0] (* Jean-François Alcover, Feb 05 2019 *)

a(n) = if(n < 3, 1, a(n\3 * 2) + 1); \\ Amiram Eldar, Jul 30 2025

a(n) = if n < 3 then 1, otherwise a(2*floor(n/3)) + 1.

a(n) = A055642(A024629(n)).

A070885 a(n) = (3/2)a(n-1) if a(n-1) is even; (3/2)(a(n-1)+1) if a(n-1) is odd.

Original entry on oeis.org

1, 3, 6, 9, 15, 24, 36, 54, 81, 123, 186, 279, 420, 630, 945, 1419, 2130, 3195, 4794, 7191, 10788, 16182, 24273, 36411, 54618, 81927, 122892, 184338, 276507, 414762, 622143, 933216, 1399824, 2099736, 3149604, 4724406, 7086609, 10629915

Offset: 1

Eric W. Weisstein, May 14 2002

nonn

The smallest positive number such that A024629(a(n)) has n digits, per page 9 of the Tanton reference in Links. - Glen Whitney, Sep 17 2017

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2002, p. 123.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
James Tanton, Exploding Dots, Chapter 9
Eric Weisstein's World of Mathematics, Wolfram Sequences

The constant K is 2/3*K(3) (see A083286). - Ralf Stephan, May 29 2003
Cf. A003312.
Cf. A081848.
Cf. A205083 (parity of terms).

a070885 n = a070885_list !! (n-1)
a070885_list = 1 : map (flip (*) 3 . flip div 2 . (+ 1)) a070885_list
-- Reinhard Zumkeller, Sep 05 2014

A070885 := proc(n)
    option remember;
    if n = 1 then
        return 1;
    elif type(procname(n-1),'even') then
        procname(n-1) ;
    else
        procname(n-1)+1 ;
    end if;
    %*3/2 ;
end proc:
seq(A070885(n),n=1..80) ; # R. J. Mathar, Jun 18 2018

NestList[If[EvenQ[#],3/2 #,3/2 (#+1)]&,1,40] (* Harvey P. Dale, May 18 2018 *)

from itertools import islice
def A070885_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a += (a+1>>1)+(a&1)
A070885_list = list(islice(A070885_gen(),70)) # Chai Wah Wu, Sep 20 2022

For n > 1, a(n) = 3*A061419(n) = 3*floor(K*(3/2)^n) where K=1.08151366859... - Benoit Cloitre, Aug 18 2002
a(n) = 3*ceiling(a(n-1)/2). - Benoit Cloitre, Apr 25 2003
a(n+1) = a(n) + A081848(n), for n > 1. - Reinhard Zumkeller, Sep 05 2014

A245356 Number of numbers whose base-4/3 expansion (see A024631) has n digits.

Original entry on oeis.org

4, 4, 4, 4, 8, 8, 12, 16, 20, 28, 36, 48, 64, 88, 116, 156, 208, 276, 368, 492, 656, 872, 1164, 1552, 2068, 2760, 3680, 4904, 6540, 8720, 11628, 15504, 20672, 27560, 36748, 48996, 65328, 87104, 116140, 154852, 206472, 275296, 367060, 489412, 652552, 870068

Offset: 1

Hailey R. Olafson, Jul 18 2014

			a(3) = 4 because 320, 321, 322, and 323 are the base-4/3 expansions for the numbers 9, 10, 11, and 12 respectively and these are the only numbers with 3 digits.

Cf. A072493, A024631, A081848.

A=[1]
for i in [1..60]:
    A.append(ceil(((4-3)/3)*sum(A)))
[4*x for x in A]

a(n) = 4*A072493(n).

A303500 The smallest positive even integer that can be written with n digits in base 3/2.

Original entry on oeis.org

2, 21, 210, 2101, 21011, 210110, 2101100, 21011000, 210110001, 2101100011, 21011000110, 210110001101, 2101100011010, 21011000110100, 210110001101001, 2101100011010011, 21011000110100110, 210110001101001101

Offset: 0

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

a(n) is a prefix of a(n+1).

The smallest, not necessarily even, integer in base 3/2 with n digits is a(n-1) with 0 added at the end.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 210 is the smallest even integer with 3 digits in base 3/2.

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

See A024629 for the base-3/2 expansion of n.

Cf. also A304024, A304025, A070885, A304272, A081848, A246435, A005428, A073941.

roll32 := proc(L)
    local piv,L1 ;
    piv := 1;
    L1 := subsop(piv=op(piv,L)+1,L) ;
    while op(piv,L1) >= 3 do
        L1 := [seq(0,i=1..piv), op(piv+1,L1)+1, seq(op(i,L1),i=piv+2..nops(L1))] ;
        piv := piv+1 ;
    end do:
    L1 ;
end proc:
from32 := proc(L)
    add( op(i,L)*(3/2)^(i-1),i=1..nops(L)) ;
end proc:
A303500 := proc(n)
    local dgs ;
    dgs := [seq(0,i=1..n-1),1] ;
    while not type(from32(dgs),'even') do
        dgs := roll32(dgs) ;
    end do:
    dgs := ListTools[Reverse](dgs) ;
    digcatL(%) ;
end proc: # R. J. Mathar, Jun 25 2018

a(n) = A024629(A305498(n)). - R. J. Mathar, Jun 25 2018

A304024 a(n) is the largest integer with n digits in base 3/2.

Original entry on oeis.org

2, 22, 212, 2122, 21222, 212212, 2122112, 21221112, 212211122, 2122111222, 21221112212, 212211122122, 2122111221212, 21221112212112, 212211122121122, 2122111221211222, 21221112212112212, 212211122121122122

Offset: 0

Tanya Khovanova and PRIMES STEP Senior group, May 04 2018

Every number starts and ends with 2 and contains only twos and ones.
Removing the last digit produces sequence A304272 of the largest even integers in base 3/2.
The value of this sequence in base 10 is A304025.
When adding 1 to the value of this sequence we get A070885.
The largest integer with a given number of digits in base 3/2 can be produced directly from the smallest number, sequence A304023, by replacing 21 at the beginning and 0 at the end with 2, and by shifting the rest up by 1, see sequence A304023.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 22 is the largest two-digit integer.

Cf. A005428, A024629, A070885, A073941, A081848, A246435, A303500, A304023, A304025, A304272.

first(n) = {my(res=vector(n), c = 2); res[1]=2; for(i=2, n, res[i] = 10 * res[i-1] + 2; if(c % 2 == 1, res[i] -= 10); c = 3 * c / 2 + if(c%2==0, 2, 1/2)); res} \\ David A. Corneth, May 11 2018

a(1) = 2, for n > 1, a(n) = 10 * a(n - 1) + 2 if A304025(n - 1) is even. Otherwise, a(n) = 10 * a(n - 1) - 8. - David A. Corneth, May 11 2018

A304025 a(n) is the largest integer that can be written with n digits in base 3/2.

Original entry on oeis.org

2, 5, 8, 14, 23, 35, 53, 80, 122, 185, 278, 419, 629, 944, 1418, 2129, 3194, 4793, 7190, 10787, 16181, 24272, 36410, 54617, 81926, 122891, 184337, 276506, 414761, 622142, 933215, 1399823, 2099735, 3149603, 4724405, 7086608, 10629914

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 04 2018

A070885 is the smallest integer that can be written with n digits in base 3/2.

This sequence represented in base 3/2 is A304024.

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore, 5 is the largest integer needing two digits in base 3/2.

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A005428, A024629, A070885, A073941, A081848, A246435, A303500, A304024, A304272.

first(n) = {my(res = vector(n)); res[1] = 2; for(i = 2, n, res[i] = 3 * res[i-1] / 2 + if(res[i-1] % 2==0, 2, 1/2));res} \\ David A. Corneth, May 11 2018

a(n) = A070885(n+1) - 1.

A304272 The largest even integer that can be written with n digits in base 3/2.

Original entry on oeis.org

2, 21, 212, 2122, 21221, 212211, 2122111, 21221112, 212211122, 2122111221, 21221112212, 212211122121, 2122111221211, 21221112212112, 212211122121122, 2122111221211221, 21221112212112212, 212211122121122121, 2122111221211221212, 21221112212112212121, 212211122121122121211, 2122111221211221212112

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

a(n) is a prefix of a(n+1).

The largest, not necessarily even, integer in base 3/2 with n digits is a(n-1) with 2 added at the end.

			The number 4 in base 3/2 is 21, and number 6 is 210. Therefore, 21 is the largest even integer with 2 digits in base 3/2.

B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A304024, A304025, A070885, A303500, A024629, A081848, A246435, A005428, A073941.

Table[StringTake["212211122121122121211221211212112", n], {n, 32}]

A304273 The concatenation of the first n terms is the smallest positive even number with n digits when written in base 3/2 (cf. A024629).

Original entry on oeis.org

2, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0

Offset: 1

Tanya Khovanova and PRIMES STEP Senior group, May 09 2018

This sequence exists since the smallest even integers (see A303500) are prefixes of each other.

Apparently a variant of A205083. - R. J. Mathar, Jun 09 2018

			The number 5 in base 3/2 is 22, and the number 6 is 210. Therefore 210 is the smallest even integer with 3 digits in base 3/2. Its prefix 21 is 4: the smallest even integer with 2 digits in base 3/2.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.

Cf. A005428, A070885, A073941, A081848, A024629, A246435, A304024, A304025, A303500, A304272, A304274.

cp's OEIS Frontend

A073941 a(n) = ceiling((Sum_{k=1..n-1} a(k)) / 2) for n >= 2 starting with a(1) = 1.

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A024629 n written in fractional base 3/2.

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A246435 Length of representation of n in fractional base 3/2.

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A070885 a(n) = (3/2)*a(n-1) if a(n-1) is even; (3/2)*(a(n-1)+1) if a(n-1) is odd.

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A245356 Number of numbers whose base-4/3 expansion (see A024631) has n digits.

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A303500 The smallest positive even integer that can be written with n digits in base 3/2.

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A304024 a(n) is the largest integer with n digits in base 3/2.

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A070885 a(n) = (3/2)a(n-1) if a(n-1) is even; (3/2)(a(n-1)+1) if a(n-1) is odd.