A108712 - OEIS

%I #28 Dec 02 2020 18:03:03

%S 1,1,2,1,3,2,4,1,5,3,6,2,7,4,8,1,9,5,1,3,0,6,1,2,1,7,1,4,2,8,1,1,3,9,

%T 1,5,4,1,1,3,5,0,1,6,6,1,1,2,7,1,1,7,8,1,1,4,9,2,2,8,0,1,2,1,1,3,2,9,

%U 2,1,2,5,3,4,2,1,4,1,2,3,5,5,2,0,6,1,2,6,7,6,2,1,8,1,2,2,9,7,3,1,0,1,3,7,1

%N A fractal sequence, defined by a(2n-1) = A007376(n) (the almost-natural numbers), a(2n) = a(n).

%C Start saying "1" and erase, as soon as they appear, the digits spelling the natural numbers. The result is the sequence itself. [The definition in the name does not yield this property - see details in the example section.]

%C Sequence based on the same skeleton as A108202 (the natural counting digits) but beginning with 1 instead of zero; with n increasing, the apparent correlation between the two sequences disappears.

%H Clark Kimberling, Un. of Evansville, <a href="http://faculty.evansville.edu/ck6/integer/fractals.html">Fractal Sequences</a>.

%F a(2n-1) = A007376(n), a(2n) = a(n).

%F a(n) = A033307(A025480(n-1)) = A007376(A025480(n-1)+1). - _Kevin Ryde_, Nov 21 2020

%e Say "1" and erase the first "1", then say "2" and erase the first "2" (leaving all other digits where they are), then say "3" and erase the first "3", etc. When it comes to "10" erase the first "1" and then the closest "0", etc. The digits to erase when the count comes to "16", for example, are next to one another. [If we apply to the sequence the process described here, the result is a different sequence, b. To get a match with the first 76 terms, we take "first" to mean "next (after the most recent erasure)". Nevertheless, we find a(76), ..., a(80) = 1,4,1,2,3; b(76), ..., b(80) = 1,1,2,4,3. - _Kevin Ryde_ and _Peter Munn_, Nov 21 2020]

%e From _Peter Munn_, Nov 21 2020: (Start)

%e Start of table showing the interleaving with the almost-natural numbers, A007376:

%e    n  a(n)  A007376    a(n/2)

%e             ((n+1)/2)

%e    1   1       1

%e    2   1                 1

%e    3   2       2

%e    4   1                 1

%e    5   3       3

%e    6   2                 2

%e    7   4       4

%e    8   1                 1

%e    9   5       5

%e   10   3                 3

%e   11   6       6

%e   12   2                 2

%e   13   7       7

%e   14   4                 4

%e   15   8       8

%e   16   1                 1

%e   17   9       9

%e   18   5                 5

%e   19   1       1

%e   20   3                 3

%e   21   0       0

%e (End)

%t f[n_] := Block[{m = 0, d = n, i = 1, l, p}, While[m <= d, l = m; m = 9i*10^(i - 1) + l; i++ ]; i--; p = Mod[d - l, i]; q = Floor[(d - l)/i] + 10^(i - 1); If[p != 0, IntegerDigits[q][[p]], Mod[q - 1, 10]]]; a[n_] := a[n] = If[EvenQ[n], a[n/2], f[(n + 1)/2]]; Table[ a[n], {n, 105}] (* _Robert G. Wilson v_, Jun 24 2005 *)

%Y Cf. A003602, A108202.

%K base,easy,nonn

%O 1,3

%A _Eric Angelini_, Jun 20 2005

%E Additional comments from _Robert G. Wilson v_ and _Alexandre Wajnberg_, Jun 24 2005

%E Incorrect formula deleted by _Peter Munn_, Nov 19 2020