fractal-like plot

[Antti Karttunen]

Santi Spadaro wrote:

> Dear seqfans,I think this curious fact deserves your attention: ID
> Number: A064770
> Sequence:  0,1,1,1,2,2,2,2,2,3,10,11,11,11,12,12,12,12,12,13,10,11,11,
>
>
> 11,12,12,12,12,12,13,10,11,11,11,12,12,12,12,12,13,20,21,21,
>
> 21,22,22,22,22,22,23,20,21,21,21,22,22,22,22,22,23,20,21,21,
>            21,22,22,22,22,22,23
> Name:      Replace each digit of n by the floor of its square root.
> Example:   26 -> [1.414...][2.449...] -> 12, so a(26) = 12.
> Keywords:  base,nonn,nice,new
> Offset:    0Author(s): Santi Spadaro (spados at katamail.com), Oct 19
> 2001 The plot of this sequence shows a curious (even if maybe not so
> surprising) fractal behaviour as you can see from the two plots
> (kindly hosted by Professor Gerard) in the links below:
>
>> http://www.seqfan.net/spadaro/A064770-1.gif
>
>> http://www.seqfan.net/spadaro/A064770-2.gif
>

Dear Santi,

Isn't this a common feature of (almost) any sequence that is generated
by
doing some transformation to _all_ digits in some base b representation
of n?
See for example, the following permutation of N:
http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=A048647



ID Number: A048647
Sequence:  0,3,2,1,12,15,14,13,8,11,10,9,4,7,6,5,48,51,50,49,60,63,62,
           61,56,59,58,57,52,55,54,53,32,35,34,33,44,47,46,45,40,43,42,
           41,36,39,38,37,16,19,18,17,28,31,30,29,24,27,26,25,20,23,22,
           21,192,195,194,193,204,207,206
Name:      Write n in base 4, then replace each digit by its base 4
negative.
Comments:  The graph of a(n) on [ 1..4^k ] resembles a plane fractal of
fractal
              dimension 1.
           Self-inverse considered as a permutation of the positive
integers.
Links:     J. W. Layman, View fractal-like graph
              URL:
http://www.math.vt.edu/people/layman/sequences/sequences.htm
           Index entries for sequences that are permutations of the
natural numbers
Example:   a(15)=5, since 15 = 33(base 4) -> 11(base 4) = 5.
See also:  Cf. A065256.
Keywords:  nonn,easy,nice
Offset:    0
Author(s): John W. Layman (layman at math.vt.edu (7/5/99))

which in turn partly inspired me to concoct this one:

ID Number: A065256
Sequence:  0,3,1,4,2,15,18,16,19,17,5,8,6,9,7,20,23,21,24,22,10,13,11,
           14,12,75,78,76,79,77,90,93,91,94,92,80,83,81,84,82,95,98,96,
           99,97,85,88,86,89,87,25,28,26,29,27,40,43,41,44,42,30,33,31,
           34,32,45,48,46,49,47,35,38
Name:      Quintal Queens permutation of N: halve or multiply by 3 (mod
5) each
              digit (0->0, 1->3, 2->1, 3->4, 4->2) of the base-5
              representation of n.
Comments:  All the permutations A004515, and A065256-A065258 consist of
the
              first fixed term ("Queen on the corner") plus infinitely
many
              4-cycles, and they satisfy the "non-attacking queen
condition" that
              p(i+d) <> p(i)+-d for all i and d >= 1.
           The corresponding infinite permutation matrix is a
scale-invariant
              fractal (Cf. A048647), and any subarray (5^i)x(5^i) (i >=
0) cut
              from its corner gives a solution to the case n=5^i of the
n nonattacking
              queens on nxn chess-board (A000170). Is there any
permutation of N
              which would give solutions to the queen problem with more
frequent
              intervals than A000351 ?
Links:     Index entries for sequences that are permutations of the
natural numbers
Maple:     [seq(QuintalQueens0Inv(j),j=0..124)];
           HalveDigit := (d,b) -> op(2,op(1,msolve(2*x=d,b))); # b
should be
              an odd integer >= 3, and d should be in range [0,b-1].
           HalveDigits := proc(n,b) local i;
              add((b^i)*HalveDigit((floor(n/(b^i)) mod
              b),b),i=0..floor(evalf(log[b](n+1)))+1); end;
           QuintalQueens0Inv := n -> HalveDigits(n,5);
See also:  Inverse permutation: A004515. A065256[n] = A065258[n+1]-1.
Cf. also
              A065187, A065189.
Keywords:  nonn,new
Offset:    0
Author(s): Antti.Karttunen at iki.fi Oct 26 2001


Terveisin,
   Ciao,

Antti


>>
>
>
>  For further explorations here's the Mathematica code: f[n_] :=
> Floor[Sqrt[n]]k[n_] := Map[f, IntegerDigits[n]]ndig[a_, b_] := 10a +
> btonum[dig_] := Fold[ndig, 0, dig]j = Table[tonum[k[n]], {n, 1,
> 100}]ListPlot[j] Change 100 in the 5th line with 10^k k=3,4,5,6,7...
> and see what happens.
>  Best Wishes,Santi Spadaro
>
> --
>