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A287730
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The s-fusc function s(n) = a(n): a(1) = 0, a(2n) = A287729(n), a(2n+1) = A287729(n) + A287729(n+1).
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10
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0, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 0, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 6, 5, 9, 4
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OFFSET
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1,7
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COMMENTS
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Define a sequence chf(n) of Christoffel words over an alphabet {-,+}:
chf(1) = '-',
chf(2*n+0) = negate(chf(n)),
chf(2*n+1) = negate(concatenate(chf(n),chf(n+1))).
Then the length of the chf(n) word is fusc(n) = A002487(n), the number of '-'-signs in the chf(n) word is c-fusc(n) = A287729(n) and the number of '+'-signs in the chf(n) word is the current sequence a(n) = s-fusc(n). See examples below.
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LINKS
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FORMULA
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The mutual diatomic recurrence pair c(n) (A287729) and s(n) (A288730) are defined by c(1)=1, s(1)=0, c(2n) = s(n), c(2n+1) = s(n)+s(n+1), s(2n) = c(n), s(2n+1) = c(n)+c(n+1).
then a(n) = k(n-1)*a(n-1) - a(n-2) - d(n) for n > 2 with a(1) = 0, a(2) = 1.
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EXAMPLE
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fusc c-fusc s-fusc
1 '-' 1 1 1 0
2 '+' 2 1 0 1
3 '+-' 2 2 1 1
4 '-' 3 1 1 0
5 '--+' 3 3 2 1
6 '-+' 3 2 1 1
7 '-++' 3 3 1 2
8 '+' 4 1 0 1
9 '+++-' 4 4 1 3
10 '++-' 4 3 1 2
11 '++-+-' 4 5 2 3
12 '+-' 4 2 1 1
13 '+-+--' 4 5 3 2
14 '+--' 4 3 2 1
15 '+---' 4 4 3 1
16 '-' 5 1 1 0
17 '----+' 5 5 4 1
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PROG
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;; An implementation of memoization-macro definec can be found for example in: http://oeis.org/wiki/Memoization
;; Second version after the alternative formula given by the author:
(Python)
from sympy.core.cache import cacheit
@cacheit
def c(n): return 1 if n==1 else s(n//2) if n%2==0 else s((n - 1)//2) + s((n + 1)//2)
@cacheit
def s(n): return 0 if n==1 else c(n//2) if n%2==0 else c((n - 1)//2) + c((n + 1)//2)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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