A007687 Number of 4-colorings of cyclic group of order n.

3, 10, 21, 44, 83, 218, 271, 692, 865, 2622, 2813, 9220, 9735, 35214, 35911, 135564, 136899, 533290, 535081

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more
• hard

Comments

The number of 2-colorings of Z_n is A000034(n-1), the number of 3-colorings of Z_n is A005843(n). The number of n-colorings of Z_2 is A137928(n-1). - Andrey Zabolotskiy, Oct 02 2017

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Robert Haas, Three-colorings of finite groups or an algebra of nonequalities, Math. Mag., 63 (1990), 211-225.
• Robert Haas, Letter to N. J. A. Sloane, Aug. 1994

Crossrefs

Cf. A007688.

Programs

Python

from itertools import product
def colorings(n, zp):
    result = 0
    for f in product(range(n), repeat=zp):
        for j1 in range(zp):
            for j2 in range(zp):
                if (f[j1]+f[j2])%n == f[(j1+j2)%zp]:
                    break
            else:
                continue
            break
        else:
            result += 1
    return result
print([colorings(4, k) for k in range(1, 12)])
# Andrey Zabolotskiy, Jul 12 2017

Extensions

a(6)-a(11) from Andrey Zabolotskiy, Jul 12 2017

a(12)-a(17) from Andrey Zabolotskiy, Oct 02 2017

a(18)-a(19) from Lucas A. Brown, Sep 20 2024