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

0, 2, 6, 12, 39, 104, 366, 1172, 4179, 14572, 52740, 190652, 700274, 2581112, 9591666, 35791472, 134236179, 505290272, 1908947406, 7233629132, 27488079132, 104715393912, 399823554006, 1529755308212, 5864066561554, 22517998136936, 86607703209516

Offset

0,2

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.

T. Pisanski, D. Schattschneider and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180. See F(n).

Formula

See Maple program.

Maple

with(numtheory):
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
H:=n-> if n mod 2 = 0 then (n/2)*4^(n/2); else 0; fi; # this is A018215 interleaved with 0's
A123045:=n-> `if`(n=0, 0, (V(n)+H(n))/(2*n));

Mathematica

V[n_] := Module[{t1 = 0}, Do[t1 = t1 + EulerPhi[k] 4^(n/k), {k, Divisors[n]}]; t1];
H[n_] := If[Mod[n, 2] == 0, (n/2) 4^(n/2), 0];
a[n_] := If[n == 0, 0, (V[n] + H[n])/(2n)];
a /@ Range[0, 26] (* Jean-François Alcover, Mar 20 2020, from Maple *)

Crossrefs

Cf. A054611, A018215.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

Sequence in context: A074442 A162589 A225957 * A280171 A327879 A094261

Adjacent sequences: A123042 A123043 A123044 * A123046 A123047 A123048

Keyword

nonn

Author

N. J. A. Sloane, Nov 11 2006

Status

approved