1,4

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).

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

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

C. Dement and Max Alekseyev, Notes on A113166

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).

(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)) ))

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

Adjacent sequences: A113163 A113164 A113165 this_sequence A113167 A113168 A113169

Sequence in context: A135291 A058617 A138135 this_sequence A126872 A094966 A095068

nonn

Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), Jan 05 2006; Jan 08 2006; Jul 29 2006

More terms from Max Alekseyev, Jun 20 2006