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 A113166 Total number of white pearls remaining in the chest - see Comments. 3
 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, 39088469, 63250315, 102334155 (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 Creighton Dement, Floretion-generated Integer Sequences (work in progress). LINKS Alois P. Heinz, Table of n, a(n) for n = 1..2000 (first 50 terms from Max Alekseyev) Creighton Dement and Max Alekseyev, Notes on A113166 FORMULA a(n) = Sum_{k=1..floor(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)) (Alekseyev). a(p) = Fibonacci(p-1) for all primes p. (Creighton Dement and Antti Karttunen, proved by Max Alekseyev). a(n) = Sum_{d|n} phi(n/d)*Fibonacci(d-1), where phi=A000010. - Maxim Karimov and Vladislav Sulima, Aug 20 2021 MAPLE with(numtheory): with(combinat): a:= n-> add(phi(d)*fibonacci(n/d-1), d=divisors(n)): seq(a(n), n=1..50);  # Alois P. Heinz, Aug 21 2021 MATHEMATICA a[n_] := Sum[EulerPhi[d]*Fibonacci[n/d - 1], {d, Divisors[n]}]; Array[a, 50] (* Jean-François Alcover, Jan 03 2022 *) 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)) )) (MATLAB) function [res] = calcA113166(n)     d=divisors(n);     res=0;     for i=1:length(d)         res=res+eulerPhi(n/d(i))*fibonacci(d(i)-1);     end end % Maxim Karimov, Aug 21 2021 CROSSREFS Cf. A000010, A034748, A006206, A000358, A000045, A000204, A000010. Sequence in context: A205977 A238623 A138135 * A126872 A336102 A094966 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 October 2 06:43 EDT 2022. Contains 357191 sequences. (Running on oeis4.)