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%I #5 Mar 14 2015 16:24:44
%S 1,3,3,8,9,20,29,60,93,189,315,618,1095,2114,3855,7414,13797,26478,
%T 49939,95838,182361,350572,671091,1292604,2485533,4797616,9256395,
%U 17903928,34636833,67125304,130150587,252677904,490853415,954502948
%N Number of different sets of n-gons labeled 1...n such that all members of each set contain equivalent paths with increasing labels; i.e., the number of isotemporal classes of n-gons.
%D B. de Bivort, Isotemporal classes of n-gons, preprint, 2004.
%D B. de Bivort, An introduction to temporal networks, preprint, 2004.
%H B. de Bivort, <a href="http://arXiv.org/abs/math.CO/0501171">Isotemporal classes of n-gons</a>
%F If n odd: (1/n) sum_{d|n} (2^(n/d-1)-1) phi(d).
%F If n = 4k + 2: (1/n) {sum_{d|n} (2^(n/d-1) phi(d)) - sum_{c|n/2} (2^(n/2c-1) phi(2c)} + 2^(n-4)/2
%F If n = 4k: (1/n) {sum_{d|n} (2^(n/d-1) phi(d)) - sum_{c|n/2} (2^(n/2c-1) phi(2c))} + 2^(n-4)/2 + 2^(n-8)/4 - 2^(ceiling[(n-4)/8]-1).
%t f[n_] := Block[{d = Divisors[n], c = Divisors[n/2]}, Switch[ Mod[n, 4], 0, (Plus @@ (2^(n/d - 1)EulerPhi[d]) - Plus @@ (2^(n/(2c) - 1)EulerPhi[2c]))/n + 2^((n - 4)/2) + 2^((n - 8)/4) - 2^(Ceiling[(n - 4)/8] - 1), 1, (Plus @@ ((2^(n/d - 1) - 1)EulerPhi[d]))/n, 2, (Plus @@ (2^(n/d - 1)EulerPhi[d]) - Plus @@ (2^(n/(2c) - 1)EulerPhi[2c]))/n + 2^((n - 4)/2), 3, (Plus @@ ((2^(n/d - 1) - 1)EulerPhi[d]))/n]]; Table[ f[n], {n, 3, 36}]
%Y Cf. A000031, A000029, A027671.
%K nonn
%O 3,2
%A Benjamin de Bivort (bivort(AT)fas.harvard.edu), Apr 03 2004
%E Edited by _Robert G. Wilson v_, Apr 09 2004
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