

A120643


Table T(n,k) = number of fractal initial sequences (where new values are successive integers) of length n whose last term is k.


0



1, 1, 1, 2, 1, 1, 3, 2, 2, 1, 5, 4, 3, 3, 1, 8, 8, 5, 6, 4, 1, 14, 14, 10, 10, 10, 5, 1, 24, 25, 21, 16, 20, 15, 6, 1, 43, 43, 43, 28, 35, 35, 21, 7, 1, 77, 76, 83, 56, 57, 70, 56, 28, 8, 1, 140, 136, 153, 120, 93, 126, 126, 84, 36, 9, 1, 256, 248, 274, 256, 165, 211, 252, 210, 120, 45, 10, 1
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OFFSET

1,4


COMMENTS

A fractal sequence is one where, when the first instance of each integer is removed, the original sequence results. We require also that these first instances occur in order: 1,1,2,3 is OK, but 1,1,3,2 is not. A finite sequence is an initial subsequence of (uncountably many) fractal sequences when the result after removing the first instance of each number is an initial subsequence. The total number of such sequences of length n is 2^{n1}. At each index after the first, the next value can be either a new value or a uniquely determined repetition of some earlier value. Conjecture: column 1 of this array is A007059.


LINKS

Table of n, a(n) for n=1..78.
C. Kimberling, Fractal sequences


FORMULA

If 2 <= n <= 2k1, T(n,k) = C(n2,k2).


EXAMPLE

For n = 3, the 4 sequences are 1,1,1; 1,1,2; 1,2,1; and 1,2,3. Of these, 2 end in 1, 1 in 2 and 1 in 3, so row 3 is 2,1,1.
The table starts:
1
1,1
2,1,1
3,2,2,1
5,4,3,3,1
8,8,5,6,4,1


MATHEMATICA

uppertrim[list_] := Fold[DeleteCases[#1, #2, 1, 1] &, list, Range[Max[list]]]; to[list_, 0] := Append[list, Part[list, Length[uppertrim@list] + 1]]; to[list_, 1] := Append[list, Max@list + 1]; allfractal[n_] := Fold[to[#1, #2] &, {1}, #] & /@ Tuples[{0, 1}, n]; k = 10; Flatten[Table[BinCounts[allfractal[k][[All, i]], {1, i + 1}] 2^(i  1), {i, k + 1}]/2^k] (* Birkas Gyorgy, Nov 25 2012 *)


CROSSREFS

Cf. A007059.
Sequence in context: A211161 A208101 A131333 * A242628 A111867 A326036
Adjacent sequences: A120640 A120641 A120642 * A120644 A120645 A120646


KEYWORD

nonn,tabl


AUTHOR

Franklin T. AdamsWatters, Aug 17 2006


STATUS

approved



