A182021 Achromatic number of n-cycle.

3, 2, 3, 3, 3, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 6, 6, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 12, 13

Offset: 3

Author

N. J. A. Sloane, Apr 06 2012

Keywords

• nonn
• easy

References

• Hare, W. R.; Hedetniemi, S. T.; Laskar, R.; Pfaff, J. Complete coloring parameters of graphs. Proceedings of the sixteenth Southeastern international conference on combinatorics, graph theory and computing (Boca Raton, Fla., 1985). Congr. Numer. 48 (1985), 171--178. MR0830709 (87h:05088)

Programs

Maple

A093353 := proc(n)
    if n < 1 then
        0;
    else
        (n + modp(n,2))*(n+1)/2 ;
    end if;
end proc:
A182021 := proc(n)
    for m from 0 do
        sm := A093353(m-1) ;
        if sm >  n then
            m := m-1 ;
            sm := A093353(m-1) ;
            if type(m,'odd') and n = sm+1 then
                return m-1 ;
            else
                return m;
            end if;
        end if;
    end do:
end proc:
seq(A182021(n),n=3..80) ; # R. J. Mathar, Jul 12 2013

Mathematica

A093353[n_] := If[n < 1, 0, (n+Mod[n, 2])*(n+1)/2];
a[n_] := For[m = 0, True, m++, sm = A093353[m-1]; If[sm > n, m = m-1; sm = A093353[m-1]; If[OddQ[m] && n == sm+1, Return[m-1], Return[m]]]];
Table[a[n], {n, 3, 80}] (* Jean-François Alcover, Apr 15 2023, after R. J. Mathar *)

Formula

Let s_m = m^2/2 if m even, m(m-1)/2 if m odd. For m >= 0, the s_m sequence is 0, 0, 2, 3, 8, 10, 18, 21, 32, 36, 50, ... (A093353 with a different offset).

Suppose s_m <= n < s_{m+1}. If m is odd and n = s_m + 1 then a(n) = m-1, otherwise a(n) = m.