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First differences of the Beatty sequence A022841 for sqrt(7).
3

%I #11 Mar 09 2019 05:16:36

%S 2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,2,

%T 3,3,2,3,3,2,3,3,2,3,3,2,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,3,2,3,2,3,3,2,

%U 3,3,2,3,3,2,3,3,2,3,3,2,3,2,3,3,2,3

%N First differences of the Beatty sequence A022841 for sqrt(7).

%C From _Michel Dekking_, Mar 09 2019: (Start)

%C This homogeneous Sturmian sequence, with the first entry removed, is fixed point of the morphism on {2,3} given by

%C 2 -> 32332332332332

%C 3 -> 32332332332332323.

%C This follows since sqrt(7)-2 has a periodic continued fraction expansion with period [1,1,1,4], see, e.g., Corollary 9.1.6 in Allouche and Shallit. (End)

%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 286.

%H Clark Kimberling, <a href="/A276857/b276857.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = floor(n*r) - floor(n*r - r), where r = sqrt(7), n >= 1.

%t z = 500; r = Sqrt[7]; b = Table[Floor[k*r], {k, 0, z}] (* A022841 *)

%t Differences[b] (* A276857 *)

%Y Cf. A022841, A276873.

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Sep 24 2016