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A113166 Total number of white pearls remaining in the chest - see Comments. 2
0, 1, 1, 3, 3, 8, 8, 17, 23, 41, 55, 102, 144, 247, 387, 631, 987, 1636, 2584, 4233, 6787, 11011, 17711, 28794, 46380, 75181, 121441, 196685, 317811, 514712, 832040, 1346921, 2178429, 3525581, 5702937, 9229314, 14930352, 24160419 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Define a(1) = 0. To calculate a(n):

1. Expand (A + B)^n into 2^n words of length n consisting of letters A and B (i.e., use of the distributive and associative laws of multiplication but assume A and B do not commute).

2. To each of the 2^n words, associate a free binary necklace consisting of n "black and white pearls". Figuratively, all 2^n necklaces can be placed inside a treasure chest.

3. Remove all n-pearled necklaces which are found to have (at least) two adjacent white pearls from the chest.

4. If two necklaces are found to be equivalent, remove one of them from the chest. Continue until no two equivalent necklaces can be found in the chest.

5. Counting the total number of white pearls left in the chest gives a(n).

REFERENCES

C. Dement, Floretion-generated Integer Sequences (work in progress).

LINKS

Max Alekseyev, Table of n, a(n) for n = 1..50

C. Dement and Max Alekseyev, Notes on A113166

FORMULA

a(n) = sum_{k=1...[n/2]} k/(n-k) sum_{j=1...gcd(n,k)} { (n-k)*gcd(n,k,j)/gcd(n,k) choose k*gcd(n,k,j)/gcd(n,k) } (Alekseyev).

a(p) = Fib(p-1) for all primes, where Fib = A000045 (Creighton Dement and Antti Karttunen, proved by Max Alekseyev).

PROG

(PARI) A113166(n) = sum(k=1, n\2, k/(n-k) * sum(j=1, gcd(n, k), binomial((n-k)*gcd([n, k, j])/gcd(n, k), k*gcd([n, k, j])/gcd(n, k)) ))

CROSSREFS

Cf. A034748, A006206, A000358, A000045, A000204.

Sequence in context: A205977 A238623 A138135 * A126872 A094966 A095068

Adjacent sequences:  A113163 A113164 A113165 * A113167 A113168 A113169

KEYWORD

nonn

AUTHOR

Creighton Dement, Jan 05 2006; Jan 08 2006; Jul 29 2006

EXTENSIONS

More terms from Max Alekseyev, Jun 20 2006

STATUS

approved

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Last modified March 29 18:39 EDT 2017. Contains 284273 sequences.