A043555 Number of runs in base-3 representation of n.

1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 3, 3, 2, 1, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 4, 3, 4, 4, 4, 3, 2, 3, 3, 2, 1, 2, 3, 3, 2, 3, 4, 4, 4, 3, 4, 3, 3, 2, 2, 3, 3, 4, 3, 4, 4, 4, 3, 3, 4, 4, 3, 2, 3, 4, 4, 3, 2, 3, 3, 3, 2, 3, 2, 2, 1, 2, 3, 3, 4, 3, 4, 4, 4, 3

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• base
• easy

Comments

Every positive integer occurs infinitely many times. See A297770 for a guide to related sequences.

Having a(0) = 1 (rather than a(0) = 0) is debatable, on the grounds that a(0) = 1 is determined by our culture, rather than the underlying mathematics. See my August 2020 comment in A145204. - Peter Munn, Jul 12 2024

From M. F. Hasler, Jul 13 2024: (Start)

The base-2 version has a(0) = 0, corresponding to the convention that 0 has zero digits, which is the more logical (but maybe less human) convention, such that, e.g., b^n is the least number with n+1 digits in base b (<=> b^n - 1 is the largest number with n digits), valid also for 0. Here and in A043556 (base 4) the convention is that 0 has one digit, '0'.

"Runs" means substrings of consecutive equal digits, here in the base-3 representation of the numbers. See Example for details. (End)

Examples

			From _M. F. Hasler_, Jul 13 2024: (Start)
Numbers n = 0, 1, 2, 3, 4, 5, ... are written '0', '1', '2', '10', '11', '12', ... in base 3. The first three have one single digit, so there is just 1 "run" (= subsequence of equal digits), whence a(0) = a(1) = a(2) = 1.
In '10' we have a "run" of '1's of length 1, followed by a run of '0's of length 1, so there are a(3) = 2 runs.
In '11' we have again one single run, here of 2 digits '1', whence a(4) = 1. (End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..19682 (first 1001 terms from Zak Seidov)

Crossrefs

Cf. A005811 (in base 2), A043556 (in base 4), A145204, A297770, A297771 (number of distinct runs).

Cf. A033113.

Programs

Maple

NRUNS := proc(L::list)
    local a,i;
    a := 1 ;
    for i from 2 to nops(L) do
        if op(i,L) <> op(i-1,L) then
            a := a+1 ;
        end if
    end do:
    a ;
end proc:
A043555 := proc(n)
    convert(n,base,3) ;
    NRUNS(%) ;
end proc:
seq(A043555(n),n=0..80) ; # R. J. Mathar, Jul 12 2024
# second Maple program:
a:= n-> `if`(n<3, 1, a(iquo(n, 3))+`if`(n mod 9 in {0, 4, 8}, 0, 1)):
seq(a(n), n=0..89);  # Alois P. Heinz, Jul 13 2024

Mathematica

b = 3; s[n_] := Length[Split[IntegerDigits[n, b]]];
Table[s[n], {n, 1, 200}]

PARI

a(n)=my(d=digits(n,3)); sum(i=2,#d,d[i]!=d[i-1])+1 \\ Charles R Greathouse IV, Jul 20 2014

Python

from itertools import groupby
from sympy.ntheory import digits
def A043555(n): return len(list(groupby(digits(n,3)[1:]))) # Chai Wah Wu, Jul 13 2024

Extensions

Updated by Clark Kimberling, Feb 03 2018