A288002 L-fusc, sequence l of the mutual diatomic recurrence pair: l(1)=0, r(1)=1, l(2n) = l(n), r(2n) = r(n), l(2n+1) = l(n)+r(n), r(2n+1) = l(n+1)+r(n+1), where r(n) = A288003(n).

0, 0, 1, 0, 1, 1, 2, 0, 1, 1, 3, 1, 2, 2, 3, 0, 1, 1, 4, 1, 3, 3, 5, 1, 2, 2, 5, 2, 3, 3, 4, 0, 1, 1, 5, 1, 4, 4, 7, 1, 3, 3, 8, 3, 5, 5, 7, 1, 2, 2, 7, 2, 5, 5, 8, 2, 3, 3, 7, 3, 4, 4, 5, 0, 1, 1, 6, 1, 5, 5, 9, 1, 4, 4, 11, 4, 7, 7, 10, 1, 3, 3, 11, 3, 8, 8

Offset

1,7

Comments

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))).

Each chf(n) word has the length fusc(n) = A002487(n) and splits uniquely into two parent Christoffel words - the left Christoffel word lef(n) of the length l-fusc(n) = a(n) and the right Christoffel word rig(n) of the length r-fusc(n) = A288003(n). See the example below.

Links

Table of n, a(n) for n=1..86.

Formula

a(n) = A002487(n) - A288003(n). [l-fusc(n) = fusc(n) - r-fusc(n).]

gcd(a(n),A288003(n)) = gcd(a(n),A002487(n)) = 1.

a(n) = A007306(n) - A287896(n).

a(n) = A007306(n) mod A002487(n).

Example

The odd bisection CHF(n) of the chf(n) sequence shifted rightwards by a(n) determines the longest overlap of the adjacent CHF words. Note that the first overlapping letters differ for n == 2^k or equivalently when a(n)==0.
To construct the word CHF(n+1) from the word CHF(n): cut off the word negate(lef(n)) of length a(n) at the left side of CHF(n), add the word negate(rig(n)) of length A288003(n) at the right side of CHF(n) and negate the first letter of the new word iff a(n)==0.
n chf(n)  A070939(n) A002487(n) lef(n) a(n)       CHF(n)
                     fusc(n)           l-fusc(n)  bisection of chf(n)
1  '-'     1          1          ''     0         '-'
2  '+'     2          1          ''     0         '+-'
3  '+-'    2          2         '+'     1         '--+'
4  '-'     3          1          ''     0          '-++'
5  '--+'   3          3         '-'     1          '+++-'
6  '-+'    3          2         '-'     1           '++-+-'
7  '-++'   3          3        '-+'     2            '+-+--'
8  '+'     4          1          ''     0              '+---'
9  '+++-'  4          4         '+'     1              '----+'
10 '++-'   4          3         '+'     1               '---+--+'
11 '++-+-' 4          5       '++-'     3                '--+--+-+'
12 '+-'    4          2         '+'     1                   '--+-+-+'
13 '+-+--' 4          5        '+-'     2                    '-+-+-++'
14 '+--'   4          3        '+-'     2                      '-+-++-++'
15 '+---'  4          4       '+--'     3                        '-++-+++'
16 '-'     5          1          ''     0                           '-++++'
17 '----+' 5          5         '-'     1                           '+++++-'

Prog

(Python)
def l(n): return 0 if n==1 else l(n//2) if n%2==0 else l((n - 1)//2) + r((n - 1)//2)
def r(n): return 1 if n==1 else r(n//2) if n%2==0 else l((n + 1)//2) + r((n + 1)//2)
print([l(n) for n in range(1, 151)]) # Indranil Ghosh, Jun 11 2017

Crossrefs

Cf. A002487, A070939, A287729, A287730, A288003.

Sequence in context: A204184 A157897 A213910 * A140129 A029347 A303427

Adjacent sequences: A287999 A288000 A288001 * A288003 A288004 A288005

Keyword

nonn

Author

I. V. Serov, Jun 10 2017

Status

approved