This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A049773 #55 Dec 08 2024 02:28:22
%S A049773 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,
%T A049773 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,
%U A049773 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
%N 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).
%C A049773 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.)
%C A049773 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 A131271. - _Colin Hall_, Jul 31 2011, Feb 29 2012
%C A049773 Row n contains one of A003407(2^(n-1)) non-averaging permutations of [2^(n-1)], i.e., a permutation of [2^(n-1)] without 3-term arithmetic progressions. - _Alois P. Heinz_, Dec 05 2017
%H A049773 Alois P. Heinz, <a href="/A049773/b049773.txt">Rows n = 1..13, flattened</a>
%H A049773 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NonaveragingSequence.html">Nonaveraging Sequence</a>.
%H A049773 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%H A049773 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bracket_%28tournament%29">Bracket (tournament)</a>.
%H A049773 <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</a>.
%e A049773 Triangle begins:
%e A049773 1;
%e A049773 1, 2;
%e A049773 1, 3, 2, 4;
%e A049773 1, 5, 3, 7, 2, 6, 4, 8;
%e A049773 1, 9, 5, 13, 3, 11, 7, 15, 2, 10, 6, 14, 4, 12, 8, 16;
%e A049773 1, 17, 9, 25, 5, 21, 13, 29, 3, 19, 11, 27, 7, 23, 15, 31, 2, 18, 10, 26, ...
%p A049773 T:= proc(n) option remember; `if`(n=1, 1,
%p A049773 [map(x->2*x-1, [T(n-1)])[], map(x->2*x, [T(n-1)])[]][])
%p A049773 end:
%p A049773 seq(T(n), n=1..7); # _Alois P. Heinz_, Oct 28 2011
%t A049773 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-François Alcover_, May 03 2012 *)
%o A049773 (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 */
%o A049773 (Haskell)
%o A049773 a049773 n k = a049773_tabf !! (n-1) !! (k-1)
%o A049773 a049773_row n = a049773_tabf !! (n-1)
%o A049773 a049773_tabf = iterate f [1] where
%o A049773 f vs = (map (subtract 1) ws) ++ ws where ws = map (* 2) vs
%o A049773 -- _Reinhard Zumkeller_, Mar 14 2015
%Y A049773 Sum of odd-indexed terms of n-th row gives A007582. Sum of even-indexed terms gives A049775.
%Y A049773 A030109 is another version.
%Y A049773 Cf. A131271.
%Y A049773 Cf. A088370.
%Y A049773 Cf. A003407, A088208.
%K A049773 nonn,tabf,nice,look
%O A049773 1,3
%A A049773 _Clark Kimberling_