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A031367
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Inflation orbit counts.
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8
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1, 0, 3, 4, 10, 12, 28, 40, 72, 110, 198, 300, 520, 812, 1350, 2160, 3570, 5688, 9348, 15000, 24444, 39402, 64078, 103320, 167750, 270920, 439128, 709800, 1149850, 1859010, 3010348, 4868640, 7880994, 12748470, 20633200, 33379200, 54018520, 87394452, 141421800
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OFFSET
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1,3
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COMMENTS
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Also number of primitive Lucas strings of length n [Ashrafi et al.] - N. J. A. Sloane, Nov 19 2014
The preceding comment is true for all n except n=2, as there are 2 primitive Lucas strings of length 2. The sequence of the number of primitive Lucas strings is the Möbius transform of the Lucas numbers A000032. - Pontus von Brömssen, Jan 24 2019
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 1..2000
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, Vertex and edge orbits of Fibonacci and Lucas cubes, 2014; See Table 3.
Michael Baake, Joachim Hermisson, Peter Pleasants, The torus parametrization of quasiperiodic LI-classes J. Phys. A 30 (1997), no. 9, 3029-3056.
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FORMULA
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If b(n) is the n-th term of A001350, then a(n) = Sum_{d|n} mu(d)b(n/d).
a(n) = n * A060280(n).
G.f.: Sum_{k>=1} mu(k) * x^k * (1 + x^(2*k)) / ((1 - x^(2*k)) * (1 - x^k - x^(2*k))). - Ilya Gutkovskiy, Feb 06 2020
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MAPLE
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A031367 := proc(n)
add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;
end proc: # R. J. Mathar, Jul 15 2016
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(binomial(a(i)/i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
end:
a:= proc(n) a(n):= ((<<0|1>, <1|1>>^n)[1, 2]-b(n, n-1))*n end:
seq(a(n), n=1..40); # Alois P. Heinz, Jun 22 2018
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MATHEMATICA
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a[n_] := n*Sum[MoebiusMu[d]*Sum[Binomial[k-1, 2k-n/d]/(n-d*k), {k, 0, n/d-1} ], {d, Divisors[n]}];
Array[a, 40] (* Jean-François Alcover, Jul 09 2018 *)
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CROSSREFS
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Cf. A001350, A006206, A000032.
Sequence in context: A101506 A092434 A239632 * A073443 A257494 A302347
Adjacent sequences: A031364 A031365 A031366 * A031368 A031369 A031370
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from James A. Sellers
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STATUS
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approved
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