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A187950
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[nr+kr] - [nr] - [kr], where r = (1+sqrt(5))/2, k = 4, [.]=floor.
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64
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1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1
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OFFSET
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1
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COMMENTS
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Suppose r is a positive irrational number and k is a positive integer, so that the sequence given by a(n) = [nr+kr] - [nr] - [kr] consists of zeros and ones (the fractional part of nr and kr being < 1). Let b(n) = (position of the n-th 0) and c(n) = (position of the n-th 1), so that b and c are a complementary pair of sequences.
Examples of a, b, c using r = (1+sqrt(5))/2:
k = 1: a = A005614 (infinite Fibonacci word),
b = A001950 (upper Wythoff sequence),
c = A000201 (lower Wythoff sequence);
Returning to arbitrary positive irrational r, let s(n) = [nr+r] - [nr] - [r], this being a(n) when k = 1. For k >= 2, the sequence a(n) = [nr+kr] - [nr] - [kr] is a shifted sum of shifted copies of s: a(n) = s(n) + s(n+1) + ... + s(n+k-1) - (constant).
It would be more natural to start the sequence with offset n = 0. -- A periodic pattern of length 21 seems to appear at n = 35 but it remains only up to n = 105. - M. F. Hasler, Oct 12 2017
This sequence is a morphic sequence, i.e., the letter-to-letter projection lambda of the fixed point of a morphism.
The fixed point is A276757=1,2,3,4,5,1,2,3,1,2,3..., the fixed point of the 4-block Fibonacci substitution on the alphabet {1,2,3,4,5} given by
1->12, 2->3, 3-> 45, 4->12, 5->3.
The letter-to-letter projection lambda is given by
1->1, 2->0, 3->1, 4->0, 5->0.
The reason that this works, is that the words of length 4 in the infinite Fibonacci word A003849 = 0100101001..., and their codings in the alphabet {1,2,3,4,5} are given by
0100 <-> 1, 1001 <-> 2, 0010 <-> 3, 0101 <-> 4, 1010 <-> 5.
The difference sequence A014755 of the lower Wythoff sequence w = A000201, given by w(n) = [n*phi] is equal to the Fibonacci word on the alphabet {2,1} (modulo a minor offset problem). This gives that the difference between [(n+4)*phi] and [n*phi] is equal to the sum w(n)+w(n+1)+w(n+2)+w(n+3), which is 6 or 7. After subtracting [4r] = floor(4*phi) = 6, these sums are equal to a(n) = lambda(A276757(n)). (End)
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LINKS
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FORMULA
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a(n) = floor(nr+4r)-floor(nr)-6, where r = (1+sqrt(5))/2.
a(n) = lambda(A276757(n)), where lambda(2) = lambda(4) = lambda(5) = 0, lambda(1) = lambda(3) = 1. - Michel Dekking, Apr 02 2020
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EXAMPLE
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a......1..0..1..1..0..1..0..1..1..0..1..1..0..1...
b......2..4..5..7..10..12.. (positions of 0 in a)
c......1..3..6..8..9..11... (positions of 1 in a).
As noted in Comments, a(n) = [nr+4r] - [nr] - [4r] is also obtained in another way: by adding left shifts of the infinite Fibonacci word s = A005614 and then down shifting:
s(n)......1..0..1..1..0..1..0..1..1..0..1..1..0..1...
s(n+1)....0..1..1..0..1..0..1..1..0..1..1..0..1..0...
s(n+2)....1..1..0..1..0..1..1..0..1..1..0..1..0..1...
s(n+3)....1..0..1..0..1..1..0..1..1..0..1..0..1..1...
sum.......3..2..3..2..2..3..2..3..3..2..3..2..2..3...
sum-2.....1..0..1..0..0..1..0..1..1..0..1..0..0..1... [Corrected by M. F. Hasler, Oct 12 2017]
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MATHEMATICA
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r = (1+5^(1/2))/2;
A187950 = Table[Floor[(n+4)r]-Floor[n*r]-6, {n, 1, 220}]
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PROG
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(Python)
from __future__ import division
from gmpy2 import isqrt
return int((isqrt(5*(n+4)**2)+n)//2 -(isqrt(5*n**2)+n)//2 - 4) # Chai Wah Wu, Oct 07 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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