A230360 a(n) is the number of base-4 n-digit numbers requiring only binary digits in bases 3 and 4.

2, 1, 0, 3, 6, 3, 0, 5, 12, 11, 0, 5, 12, 0, 0, 5, 0, 0, 0, 48, 14, 0, 61, 188, 83, 0, 81, 232, 268, 0, 0, 650, 0, 0, 622, 299, 0, 0, 0, 501, 0, 0, 2655, 602, 0, 6429, 8990, 7856, 0, 26187, 17898, 3744, 0, 40300, 16395, 0, 0, 0, 0, 0, 0, 124876, 173552, 0, 0

Offset

1,1

Comments

0 is included to mesh with the earlier A146025 ({0, 1, 82000}: the values in decimal with top base 4 here replaced by 5, conjectured complete). It appears unlikely, empirically, that this sequence has a last positive term, and a heuristic approximation of terms is likely not difficult.

The numbers here are the counts of members of A000695 also occurring in A005836 and being n digits in length in base-4.

Links

Table of n, a(n) for n=1..65.

Stuart A. Burrell, Han Yu, Digit expansions of numbers in different bases, arXiv:1905.00832 [math.NT], 2019.

Example

The first 8 values are 0, 1, 4, 81, 84, 85, 256 and 273--0, 1, 11, 10000, 10010, 10011, 10000111 and 10001010 in base 3, and 0, 1, 10, 1101, 1110, 1111, 10000 and 10101 in base 4; and, from the base-4 listing, a(1)=2, a(2)=1, a(3)=0, a(4)=3, and a(5) is at least 2.

Mathematica

MapAt[# + 1 &, Array[Count[FromDigits[#, 4] & /@ IntegerDigits[Range[2^(# - 1), 2^# - 1], 2], _?(DigitCount[#, 3][[2]] == 0 &)] &, 20], 1] (* Michael De Vlieger, Jun 11 2019 *)

Prog

(PARI)
{
\\ This program finds the number of d-digit base B>b\\
\\ numbers not requiring digits beyond those of base b\\
\\ for bases b+1 through B. It runs a check in reverse\\
\\ down to base b+1, maintaining additions not yet done\\
\\ in vector S, where the digits in each base are kept\\
\\ in matrix N.  The value itself is kept as n, at each\\
\\ new base checked for n, the value in S is transfered\\
\\ to variable t; with the check being done of whether\\
\\ the criterion is satisfied for n in the base under\\
\\ consideration.  A flag f is used to see if n passed\\
\\ for all bases or there was a break.  If pass, then\\
\\ count variable c is incremented (as is n) for the\\
\\ next run through bases.  At each addition, a check\\
\\ of whether the base is B and the number of digits\\
\\ changes is done, and if so a new term is output.\\
\\ pos and POS are variables for the digit-positions\\
\\ under consideration in additions essentially mimic-\\
\\ king hand addition.  Flag g identifies whether or\\
\\ not a large addition is warranted by virtue of an\\
\\ addition resulting in a digit larger than b-1, the\\
\\ leftmost of these being the point from which this\\
\\ addition is made using variable s calculated four\\
\\ lines above the bottom one of the program.  This \\
\\ program is readily modified  to store a smaller #\\
\\ of digits (D), change the b and B values, and print\\
\\ specific n values as desired.\\
b=2; B=4; d=1; c=1; D=10000;
N=matrix(B-b, D); n=1; S=vector(B-b, x, 1);
while(1,
  f=1; forstep(i=B, b+1, -1,
    t=S[i-b]; if(t,
      S[i-b]=0; pos=0; ca=0;
      while(t,
        pos++; N[i-b, pos]+=t%i+ca;
        if(N[i-b, pos]>=i, ca=1; N[i-b, pos]-=i, ca=0);
        t\=i);
      if(ca, pos++; N[i-b, pos]++; if(i==B, if(pos==d+1,
            print1(c", "); d++; c=0)));
      POS=pos; g=1;
      while(POS,
        if(N[i-b, POS]>=b, g=0; break(), POS--));
      if(g==0,
        f=0; POS++; while(N[i-b, POS]==b-1, POS++);
        N[i-b, POS]++; for(j=1, POS-1, N[i-b, j]=0);
        s=i^(POS-1)-n%(i^(POS-1));
        for(j=1, B-b, if(j!=i-b, S[j]+=s));
        if(i==B, if(POS==d+1, print1(c", "); d++; c=0));
        n+=s; break())));
  if(f, c++; n++; S=vector(B-b, x, 1)))
}

Crossrefs

Cf. A146025, A005836, A000695

Sequence in context: A267222 A077874 A286274 * A244117 A263426 A357103

Adjacent sequences: A230357 A230358 A230359 * A230361 A230362 A230363

Keyword

nonn,base,uned

Author

James G. Merickel, Oct 16 2013

Status

approved