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A080764
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First differences of A049472, floor(n/sqrt(2)).
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40
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1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0
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OFFSET
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0,1
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COMMENTS
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Fixed point of the morphism 0->1, 1->110. - Benoit Cloitre, May 31 2004
As binary constant 0.1101101110110... = 0.85826765646... (A119812), see Fxtbook link. - Joerg Arndt, May 15 2011
Characteristic word with slope 1/sqrt(2) [see J. L. Ramirez et al.]. - R. J. Mathar, Jul 09 2013
Sturmian word: equals the limit word S(infinity) where S(0) = 0, S(1) = 1 and for n >= 1, S(n+1) = S(n)S(n)S(n-1).
More generally, for k = 0,1,2,..., we can define a sequence of words S_k(n) by S_k(0) = 0, S_k(1) = 0...01 (k 0's) and for n >= 1, S_k(n+1) = S_k(n)S_k(n)S_k(n-1). Then the limit word S_k(infinity) is a Sturmian word whose terms are given by a(n) = floor((n + 2)/(k + sqrt(2))) - floor((n + 1)/(k + sqrt(2))).
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LINKS
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FORMULA
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a(n) = floor((n+2)*sqrt(2)/2) - floor((n+1)*sqrt(2)/2).
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EXAMPLE
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The first few Sturmian words S(n) are
S(0) = 0
S(1) = 1
S(2) = 110
S(3) = 110 110 1
S(4) = 1101101 1101101 110
S(5) = 11011011101101110 11011011101101110 1101101
The lengths of the words are [1, 1, 3, 7, 17, 41, ...] = A001333. (End)
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MAPLE
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alpha := 1/sqrt(2) ;
floor((n+2)*alpha)-floor((n+1)*alpha) ;
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MATHEMATICA
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Nest[ Flatten[ # /. {0 -> 1, 1 -> {1, 1, 0}}] &, {1}, 7] (* Robert G. Wilson v, Apr 16 2005 *)
NestList[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0, 1}}] &, {1}, 5] // Flatten (* or *)
t = Table[Floor[n/Sqrt[2]], {n, 111}]; Drop[t, 1] - Drop[t, -1] (* Robert G. Wilson v, Nov 03 2005 *)
a[ n_] := With[{m = n + 1}, Floor[(m + 1) / Sqrt[2]] - Floor[m / Sqrt[2]]]; (* Michael Somos, Aug 19 2018 *)
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PROG
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(Haskell)
a080764 n = a080764_list !! n
a080764_list = tail $ zipWith (-) (tail a049472_list) a049472_list
(PARI) {a(n) = n++; my(k = sqrtint(n*n\2)); n*(n+2) > 2*k*(k+2)}; /* Michael Somos, Aug 19 2018 */
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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