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A031367 Inflation orbit counts.

%I A031367 #38 Jul 02 2025 16:01:56
%S A031367 1,0,3,4,10,12,28,40,72,110,198,300,520,812,1350,2160,3570,5688,9348,
%T A031367 15000,24444,39402,64078,103320,167750,270920,439128,709800,1149850,
%U A031367 1859010,3010348,4868640,7880994,12748470,20633200,33379200,54018520,87394452,141421800
%N A031367 Inflation orbit counts.
%C A031367 Also number of primitive Lucas strings of length n [Ashrafi et al.] - _N. J. A. Sloane_, Nov 19 2014
%C A031367 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
%H A031367 Alois P. Heinz, <a href="/A031367/b031367.txt">Table of n, a(n) for n = 1..2000</a>
%H A031367 A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-August-11-2014.pdf">Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings</a>, 2014.
%H A031367 A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar and M. Petkovsek, <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-submit-2014.pdf">Vertex and edge orbits of Fibonacci and Lucas cubes</a>, 2014; See Table 3.
%H A031367 Michael Baake, Joachim Hermisson, Peter Pleasants, <a href="http://dx.doi.org/10.1088/0305-4470/30/9/016">The torus parametrization of quasiperiodic LI-classes</a> J. Phys. A 30 (1997), no. 9, 3029-3056.
%F A031367 If b(n) is the n-th term of A001350, then a(n) = Sum_{d|n} mu(d)b(n/d).
%F A031367 a(n) = n * A060280(n).
%F A031367 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
%p A031367 A031367 := proc(n)
%p A031367     add( numtheory[mobius](d)*A001350(n/d), d=numtheory[divisors](n)) ;
%p A031367 end proc: # _R. J. Mathar_, Jul 15 2016
%p A031367 # second Maple program:
%p A031367 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A031367       add(binomial(a(i)/i+j-1, j)*b(n-i*j, i-1), j=0..n/i)))
%p A031367     end:
%p A031367 a:= proc(n) a(n):= ((<<0|1>, <1|1>>^n)[1, 2]-b(n, n-1))*n end:
%p A031367 seq(a(n), n=1..40);  # _Alois P. Heinz_, Jun 22 2018
%t A031367 a[n_] := n*Sum[MoebiusMu[d]*Sum[Binomial[k-1, 2k-n/d]/(n-d*k), {k, 0, n/d-1} ], {d, Divisors[n]}];
%t A031367 Array[a, 40] (* _Jean-François Alcover_, Jul 09 2018 *)
%Y A031367 Cf. A001350, A006206, A000032.
%K A031367 nonn,easy
%O A031367 1,3
%A A031367 _N. J. A. Sloane_
%E A031367 More terms from _James Sellers_