A092481 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.

1, 3, 3, 8, 9, 20, 29, 60, 93, 189, 315, 618, 1095, 2114, 3855, 7414, 13797, 26478, 49939, 95838, 182361, 350572, 671091, 1292604, 2485533, 4797616, 9256395, 17903928, 34636833, 67125304, 130150587, 252677904, 490853415, 954502948

Offset: 3

Author

Benjamin de Bivort (bivort(AT)fas.harvard.edu), Apr 03 2004

Keywords

• nonn

References

• B. de Bivort, Isotemporal classes of n-gons, preprint, 2004.
• B. de Bivort, An introduction to temporal networks, preprint, 2004.

Links

• B. de Bivort, Isotemporal classes of n-gons

Crossrefs

Cf. A000031, A000029, A027671.

Programs

Mathematica

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}]

Formula

If n odd: (1/n) sum_{d|n} (2^(n/d-1)-1) phi(d).

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

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).

Extensions

Edited by Robert G. Wilson v, Apr 09 2004