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A049773 Triangular array T read by rows: if row n is r(1),...,r(m), then row n+1 is 2r(1)-1,...,2r(m)-1,2r(1),...,2r(m). 10
1, 1, 2, 1, 3, 2, 4, 1, 5, 3, 7, 2, 6, 4, 8, 1, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6, 14, 4, 12, 8, 16, 1, 17, 9, 25, 5, 21, 13, 29, 3, 19, 11, 27, 7, 23, 15, 31, 2, 18, 10, 26, 6, 22, 14, 30, 4, 20, 12, 28, 8, 24, 16, 32, 1, 33, 17, 49, 9, 41, 25, 57, 5, 37, 21, 53, 13, 45, 29, 61, 3, 35, 19 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

n-th row = (r(1),r(2),...,r(m)), where m=2^(n-1), satisfies r(r(k))=k for k=1,2,...,m and has exactly 2^[ n/2 ] solutions of r(k)=k. (The function r(k) reverses bits. Or rather, r(k)=revbits(k-1)+1.

In a knockout competition with m players, arranging the competition brackets (see links) in r(k) order, where k is the rank of the player, ensures that highest ranked players cannot meet until the later stages of the competition.  None of the top 2^p ranked players can meet earlier than the p-th from last round of the competition.  At the same time the top ranked players in each match meet the highest ranked player possible consistent with this rule.  The sequence for the top ranked players meeting the lowest ranked player possible is A208569.  See also A131271. - Colin Hall, Jul 31 2011, Feb 29 2012

LINKS

Alois P. Heinz, Rows n = 1..13, flattened

Wikipedia, Bracket (tournament)

EXAMPLE

Rows: {1}; {1,2}; {1,3,2,4}; {1,5,3,7,2,6,4,8}; ...

MAPLE

T:= proc(n) option remember; `if`(n=1, 1,

      [map(x->2*x-1, [T(n-1)])[], map(x->2*x, [T(n-1)])[]][])

    end:

seq(T(n), n=1..7);  # Alois P. Heinz, Oct 28 2011

MATHEMATICA

row[1] = {1}; row[n_] := row[n] = Join[ 2*row[n-1] - 1, 2*row[n-1] ]; Flatten[ Table[ row[n], {n, 1, 7}]] (* Jean-Fran├žois Alcover, May 03 2012 *)

PROG

(PARI) (a(n, k) = if( k<=0 || k>=n, 0, if( k%2, n\2) + a(n\2, k\2))); {T(n, k) = if( k<=0 || k>2^n/2, 0, 1 + a(2^n/2, k-1))}; /* Michael Somos, Oct 13 1999 */

(Haskell)

a049773 n k = a049773_tabf !! (n-1) !! (k-1)

a049773_row n = a049773_tabf !! (n-1)

a049773_tabf = iterate f [1] where

   f vs = (map (subtract 1) ws) ++ ws where ws = map (* 2) vs

-- Reinhard Zumkeller, Mar 14 2015

CROSSREFS

Sum of odd-indexed terms of n-th row gives A007582. Sum of even-indexed terms gives A049775.

A030109 is another version.

A131271.

Cf. A088370. - Alois P. Heinz, Oct 28 2011

Cf. A208569. - Colin Hall, Feb 29 2012

Cf. A088208.

Sequence in context: A003602 A265650 A181733 * A261401 A217669 A123021

Adjacent sequences:  A049770 A049771 A049772 * A049774 A049775 A049776

KEYWORD

nonn,tabf,nice,look

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified August 16 19:44 EDT 2017. Contains 290627 sequences.