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Binary array, related to the Thue-Morse sequence A010060, read by downward antidiagonals.
3

%I #19 Apr 04 2024 10:04:59

%S 0,1,0,0,1,0,1,1,1,0,0,0,1,1,0,1,0,0,1,1,0,0,1,1,0,1,1,0,1,1,0,1,0,1,

%T 1,0,0,0,0,0,1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,0,1,1,

%U 1,1,1,0,0,1,0,1,1,0,0,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0,0,1,1,0,0,1,0,1,1

%N Binary array, related to the Thue-Morse sequence A010060, read by downward antidiagonals.

%H Paolo Xausa, <a href="/A104894/b104894.txt">Table of n, a(n) for n = 0..11324</a> (first 150 antidiagonals, flattened).

%H David Applegate, Benoit Cloitre, Philippe Deléham and N. J. A. Sloane, Sloping binary numbers: a new sequence related to the binary numbers [<a href="http://neilsloane.com/doc/slopey.pdf">pdf</a>, <a href="http://neilsloane.com/doc/slopey.ps">ps</a>].

%F T(n,k) = A010060(k mod 2^(n+1)). - _Paolo Xausa_, Apr 04 2024

%e Row n consists of repetitions of the first 2^(n+1) terms of A010060:

%e .

%e n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...

%e ---------------------------------------------------------

%e 0 | 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...

%e 1 | 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, ...

%e 2 | 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, ...

%e 3 | 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...

%e 4 | 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...

%e 5 | 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...

%e 6 | 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...

%e 7 | 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, ...

%e ...

%e Columns written in base 10 (see Table 2 of reference) give 0, -1, -2, 1, -4, 3, 2, -3, -8, 7, 6, -7, 4, -5, -6, 5, ... (see A104895).

%t Table[ThueMorse[Mod[n-k, 2^(k+1)]], {n, 0, 15}, {k, 0, n}] (* _Paolo Xausa_, Apr 04 2024 *)

%Y Cf. A010060, A104895.

%K nonn,tabl

%O 0,1

%A _Philippe Deléham_, Apr 24 2005

%E a(48) = 0 removed by _Paolo Xausa_, Apr 04 2024