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A182021 Achromatic number of n-cycle.

%I A182021 #13 Apr 15 2023 05:04:16
%S A182021 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,
%T A182021 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,
%U A182021 11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,12,13
%N A182021 Achromatic number of n-cycle.
%D A182021 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)
%F A182021 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).
%F A182021 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.
%p A182021 A093353 := proc(n)
%p A182021     if n < 1 then
%p A182021         0;
%p A182021     else
%p A182021         (n + modp(n,2))*(n+1)/2 ;
%p A182021     end if;
%p A182021 end proc:
%p A182021 A182021 := proc(n)
%p A182021     for m from 0 do
%p A182021         sm := A093353(m-1) ;
%p A182021         if sm >  n then
%p A182021             m := m-1 ;
%p A182021             sm := A093353(m-1) ;
%p A182021             if type(m,'odd') and n = sm+1 then
%p A182021                 return m-1 ;
%p A182021             else
%p A182021                 return m;
%p A182021             end if;
%p A182021         end if;
%p A182021     end do:
%p A182021 end proc:
%p A182021 seq(A182021(n),n=3..80) ; # _R. J. Mathar_, Jul 12 2013
%t A182021 A093353[n_] := If[n < 1, 0, (n+Mod[n, 2])*(n+1)/2];
%t A182021 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]]]];
%t A182021 Table[a[n], {n, 3, 80}] (* _Jean-François Alcover_, Apr 15 2023, after _R. J. Mathar_ *)
%K A182021 nonn,easy
%O A182021 3,1
%A A182021 _N. J. A. Sloane_, Apr 06 2012