A080764 First differences of A049472, floor(n/sqrt(2)).

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

Offset: 0

Author

Matthew Vandermast, Mar 25 2003

Keywords

• nonn
• easy

Comments

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

From Peter Bala, Nov 22 2013: (Start)

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))).

This sequence corresponds to the case k = 0. See A159684 (case k = 1) and A171588 (case k = 2). Compare with the Fibonacci words A005614, A221150, A221151 and A221152. See also A230901. (End)

For n > 0: a(A001951(n)) = 1, a(A001952(n)) = 0. - Reinhard Zumkeller, Jul 03 2015

Binary complement of the Pell word A171588. - Michel Dekking, Feb 22 2018

Examples

			From _Peter Bala_, Nov 22 2013: (Start)
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)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Joerg Arndt, Matters Computational (The Fxtbook), section 38.12, pp. 757-758.
• Wikipedia, Sturmian word
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A005614, A159684, A171588, A221150, A221151, A221152, A230901.

Cf. A049472, A001951, A001952, A188295.

Programs

Haskell

a080764 n = a080764_list !! n
a080764_list = tail $ zipWith (-) (tail a049472_list) a049472_list
-- Reinhard Zumkeller, Jul 03 2015

Maple

A080764 := proc(n)
    alpha := 1/sqrt(2) ;
    floor((n+2)*alpha)-floor((n+1)*alpha) ;
end proc: # R. J. Mathar, Jul 09 2013

Mathematica

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 *)

PARI

{a(n) = n++; my(k = sqrtint(n*n\2)); n*(n+2) > 2*k*(k+2)}; /* Michael Somos, Aug 19 2018 */

Python

from math import isqrt
def A080764(n): return (isqrt((m:=(n+2)**2)<<1)>>1)-(isqrt(m-(n<<1)-3<<1)>>1) # Chai Wah Wu, May 19 2025

Formula

a(n) = floor((n+2)*sqrt(2)/2) - floor((n+1)*sqrt(2)/2).

a(n) = A188295(n+2) for all n in Z. - Michael Somos, Aug 19 2018