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A088370 Triangle T(n,k), read by rows, where the n-th row is a binary arrangement of the numbers 1 through n. 5
1, 1, 2, 1, 3, 2, 1, 3, 2, 4, 1, 5, 3, 2, 4, 1, 5, 3, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 1, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 3, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 8, 1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 4, 12, 8, 1, 9, 5, 13, 3, 11, 7, 2 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The n-th row differs from the prior row only by the presence of n. See A088371 for the positions in the n-th row that n is inserted.

Comment from Clark Kimberling, Aug 02 2007: (Start)

At A131966, this sequence is cited as the fractal sequence of the Cantor set C.

Recall that C is the set of fractions in [0,1] whose base 3 representation consists solely of 0s and 2s.

Arrange these fractions as follows:

0

0, .2

0, .02, .2

0, .02, .2, .22

0, .002, .02, .2, .22, etc.

Replace each number x by its order of appearance, counting each distinct predecessor of x only once, getting

1;

1, 2;

1, 3, 2;

1, 3, 2, 4;

1, 5, 3, 2, 4;

Concatenate these to get the current sequence, which is a fractal sequence as defined in "Fractal sequences and interspersions".

One property of such a sequence is that it properly contains itself as a subsequence (infinitely many times). (End)

REFERENCES

Clark Kimberling, "Fractal sequences and interspersions," Ars Combinatoria 45 (1997) 157-168.

LINKS

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

T(n,n) = 2^(floor(log(n)/log(2))). Construction. The 2n-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n, after multiplying each term by 2. The (2n-1)-th row is the concatenation of row n, after multiplying each term by 2 and subtracting 1, with row n-1, after multiplying each term by 2.

EXAMPLE

Row 5 is formed from row 3, {1,3,2} and row 2, {1,2}, like so:

{1,5,3, 2,4} = {1*2-1, 3*2-1, 2*2-1} | {1*2, 2*2}.

Triangle begins:

1;

1, 2;

1, 3, 2;

1, 3, 2, 4;

1, 5, 3, 2, 4;

1, 5, 3, 2, 6, 4;

1, 5, 3, 7, 2, 6, 4;

1, 5, 3, 7, 2, 6, 4, 8;

1, 9, 5, 3, 7, 2, 6, 4, 8;

1, 9, 5, 3, 7, 2, 10, 6, 4, 8;

1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 8;

1, 9, 5, 3, 11, 7, 2, 10, 6, 4, 12, 8;

1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 4, 12, 8;

1, 9, 5, 13, 3, 11, 7, 2, 10, 6, 14, 4, 12, 8;

1, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6, 14, 4, 12, 8;

1, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6, 14, 4, 12, 8, 16;

1, 17, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6, 14, 4, 12, 8, 16; ...

MAPLE

T:= proc(n) option remember;

      `if`(n=1, 1, [map(x-> 2*x-1, [T(n-iquo(n, 2))])[],

                    map(x-> 2*x,   [T(  iquo(n, 2))])[]][])

    end:

seq(T(n), n=1..20);  # Alois P. Heinz, Oct 28 2011

MATHEMATICA

T[1] = {1}; T[n_] := T[n] = Join[q = Quotient[n, 2]; 2*T[n-q]-1, 2*T[q]]; Table[ T[n], {n, 1, 20}] // Flatten (* Jean-Fran├žois Alcover, Feb 26 2015, after Alois P. Heinz *)

PROG

(PARI) {T(n, k) = if(k==0, 1, if(k<=n\2, 2*T(n\2, k) - 1, 2*T((n-1)\2, k-1-n\2) ))}

for(n=0, 20, for(k=0, n, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A088371.

Diagonal gives A053644. Cf. A049773. - Alois P. Heinz, Oct 28 2011

Sequence in context: A082074 A132283 A256440 * A113787 A115624 A076291

Adjacent sequences:  A088367 A088368 A088369 * A088371 A088372 A088373

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Sep 28 2003

STATUS

approved

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Last modified June 28 17:15 EDT 2017. Contains 288839 sequences.