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A276633 a(n) = smallest integer not yet in the sequence with no digits in common with a(n-1) and a(n-2); a(0)=0, a(1)=1. 8
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 33, 11, 20, 34, 15, 26, 30, 14, 25, 36, 17, 24, 35, 16, 27, 38, 19, 40, 23, 18, 44, 29, 13, 45, 28, 31, 46, 50, 12, 37, 48, 21, 39, 47, 51, 32, 49, 55, 60, 41, 52, 63, 70, 42, 53, 61, 72, 43, 56, 71, 80, 54, 62, 73, 58, 64, 77, 59, 66, 74, 81, 65, 79 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sequence is not a permutation of the positive integers. E.g., 123456789 and 1023456789 (the smallest pandigital number) are not members.

Numbers n such that a(n)=n: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 52, 147, 1619, 6140, ...

The sequence is infinite, since all digits in a(n-3) are allowed in a(n). - Robert Israel, Sep 20 2016

LINKS

Zak Seidov and David A. Corneth, Table of n, a(n) for n = 0..19999 (First 2001 terms from Zak Seidov).

EXAMPLE

From David A. Corneth, Sep 22 2016: (Start) Each number can consist of 2^10-1 sets of distinct digits, i.e., classes. For example, 21132 is in the class {1, 2, 3}. We don't include a number without digits. For this sequence, we can also exclude numbers with only the digit 0. This leaves 1022 classes. We create a list with a place for each class containing the least number from that class not already in the sequence.

To illustrate the algorithm used to create the current b-file, we'll (for brevity) assume we've already calculated all terms for n = 1 to 100 and that we already know which classes will be used to compute the next 10 terms, for n = 101 to 110.

These classes are:  {0, 1}, {2, 3}, {5, 9}, {7, 9}, {8, 9}, {0, 1, 6}, {0, 1, 7}, {2, 2, 2} and {2, 2, 4} having the values 110, 223, 95, 97, 89, 106, 107, 222 and 224. a(99) = 104 and a(100) = 88, so from those values we may only choose from {223, 95, 97 and 222}. The least value in the list is 95. Therefore, a(101) = 95. The number for the class is now replaced with the next larger number having digits {5, 9} (=A276769(95)), being 559.

(One may see that in the example I only listed 9 classes. Class {8, 9} occurs twice in the example; a(104) = 89 and a(107) = 98.)

From a list of computed values up to some n, the values for classes may be updated to compute further. E.g., to compute a(20000), one may use the b-file to find the least number not already in the sequence for each class and then proceed from a(19998) and a(19999), etc. (End)

MAPLE

N:= 10^3: # to get all terms before the first > N

for R in combinat:-powerset({$0..9}) minus {{}, {$0..9}} do

  Lastused[R]:= [];

  MR[R]:= Array[0..9];

  for i from 1 to nops(R) do MR[R][R[i]]:= i od:

od:

A[0]:= 0: A[1]:= 1:

S:= {0, 1}:

for n from 2 to N do

  R:= {$0..9} minus (convert(convert(A[n-1], base, 10), set) union convert(convert(A[n-2], base, 10), set));

  L:= Lastused[R];

  x:= 0;

  while member(x, S) do

    for d from 1 do

      if d > nops(L) then

        if R[1] = 0 then L:= [op(L), R[2]] else L:= [op(L), R[1]] fi;

        break

      elif L[d] < R[-1] then

        L[d]:= R[MR[R][L[d]]+1]; break

      else

        L[d]:= R[1];

      fi

    od;

    x:= add(L[j]*10^(j-1), j=1..nops(L));

  od;

  A[n]:= x;

  S:= S union {x};

  Lastused[R] := L;

od:

seq(A[i], i=0..N); # Robert Israel, Sep 20 2016

MATHEMATICA

s={0, 1}; Do[a=s[[-2]]; b=s[[-1]]; n=2; idab=Union[IntegerDigits[a], IntegerDigits[b]]; While[MemberQ[s, n]|| Intersection[idab, IntegerDigits[n]]!={}, n++]; AppendTo[s, n], {100}]; s

CROSSREFS

Cf. A067581, A086066, A276512, A276769.

Sequence in context: A116069 A081511 A030283 * A298482 A301801 A308540

Adjacent sequences:  A276630 A276631 A276632 * A276634 A276635 A276636

KEYWORD

nonn,base,easy

AUTHOR

Zak Seidov and Eric Angelini, Sep 08 2016

STATUS

approved

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Last modified August 21 06:54 EDT 2019. Contains 326162 sequences. (Running on oeis4.)