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A123045 Number of frieze patterns of length n under a certain group (see Pisanski et al. for precise definition).

%I A123045 #20 Mar 20 2020 10:06:00
%S A123045 0,2,6,12,39,104,366,1172,4179,14572,52740,190652,700274,2581112,
%T A123045 9591666,35791472,134236179,505290272,1908947406,7233629132,
%U A123045 27488079132,104715393912,399823554006,1529755308212,5864066561554,22517998136936,86607703209516
%N A123045 Number of frieze patterns of length n under a certain group (see Pisanski et al. for precise definition).
%H A123045 Alois P. Heinz, <a href="/A123045/b123045.txt">Table of n, a(n) for n = 0..1000</a>
%H A123045 Shinsaku Fujita, <a href="https://www.webfile.jp/lite-tcsj/dl.php?i=2071&amp;s=aaa3068608b595219b60">alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method</a>, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
%H A123045 T. Pisanski, D. Schattschneider and B. Servatius, <a href="http://www.jstor.org/stable/27642932">Applying Burnside's lemma to a one-dimensional Escher problem</a>, Math. Mag., 79 (2006), 167-180. See F(n).
%F A123045 See Maple program.
%p A123045 with(numtheory):
%p A123045 V:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*4^(n/k); od: t1; end; # A054611
%p A123045 H:=n-> if n mod 2 = 0 then (n/2)*4^(n/2); else 0; fi; # this is A018215 interleaved with 0's
%p A123045 A123045:=n-> `if`(n=0,0, (V(n)+H(n))/(2*n));
%t A123045 V[n_] := Module[{t1 = 0}, Do[t1 = t1 + EulerPhi[k] 4^(n/k), {k, Divisors[n]}]; t1];
%t A123045 H[n_] := If[Mod[n, 2] == 0, (n/2) 4^(n/2), 0];
%t A123045 a[n_] := If[n == 0, 0, (V[n] + H[n])/(2n)];
%t A123045 a /@ Range[0, 26] (* _Jean-François Alcover_, Mar 20 2020, from Maple *)
%Y A123045 Cf. A054611, A018215.
%Y A123045 The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.
%K A123045 nonn
%O A123045 0,2
%A A123045 _N. J. A. Sloane_, Nov 11 2006