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A127687
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Number of unlabeled maximal independent sets in the n-cycle graph.
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4
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0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 5, 6, 7, 7, 11, 11, 16, 19, 24, 28, 39, 46, 60, 75, 97, 120, 159, 197, 257, 327, 422, 539, 700, 892, 1157, 1488, 1928, 2479, 3219, 4148, 5383, 6961, 9029, 11687, 15184, 19673, 25564, 33174, 43125, 56010, 72868, 94719, 123283
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OFFSET
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1,6
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COMMENTS
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Number of unlabeled (i.e., defined up to a rotation) maximal independent sets in the n-cycle graph. Also: Number of cyclic compositions of n in which each term is either 2 or 3.
(a(n)*n - A001608(n)) mod 2 is a binary sequence of period 14: [0,0,0,0,0,1,0,0,0,1,0,1,0,1]. - Richard Turk, Aug 25 2017
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LINKS
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FORMULA
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G.f.: Sum_{k>=1} (phi(k)/k) * log( 1/(1-B(x^k)) ) where B(x) = x^2 + x^3. - Joerg Arndt, Jan 21 2013
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MAPLE
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with(numtheory):
perrin:=proc(n) option remember: if n=0 then 3 elif n=1 then 0 elif n=2 then 2 else perrin(n-2)+perrin(n-3) fi end:
a:=proc(n) local d, N:d:=divisors(n); N:=nops(d):
add(phi(n/d[k])*perrin(d[k]), k=1..N)/n end:
seq(a(n), n=1..50);
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MATHEMATICA
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(* p = A001608 *) p[n_] := p[n] = p[n - 2] + p[n - 3]; p[0] = 3; p[1] = 0; p[2] = 2; A113788[n_] := (1/n)*Sum[MoebiusMu[n/d]*p[d], {d, Divisors[n]}]; a[n_] := Sum[A113788[d], {d, Divisors[n]}]; Table[a[n], {n, 1, 56}] (* Jean-François Alcover, Jul 16 2012, from formula *)
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PROG
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(PARI)
N = 66; x = 'x + O('x^N);
f(x) = x^2+x^3;
gf = 'c0 + sum(n=1, N, eulerphi(n)/n*log(1/(1-f(x^n))) );
v = Vec(gf); v[1]-='c0; v /* includes a(0)=0 */
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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Jean-Luc Marichal (jean-luc.marichal(AT)uni.lu), Jan 24 2007
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STATUS
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approved
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