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Achromatic number of n-cycle.
1

%I #13 Apr 15 2023 05:04:16

%S 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 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 11,11,11,11,11,11,11,11,12,12,12,12,12,12,13,12,13

%N Achromatic number of n-cycle.

%D 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 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 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 A093353 := proc(n)

%p if n < 1 then

%p 0;

%p else

%p (n + modp(n,2))*(n+1)/2 ;

%p end if;

%p end proc:

%p A182021 := proc(n)

%p for m from 0 do

%p sm := A093353(m-1) ;

%p if sm > n then

%p m := m-1 ;

%p sm := A093353(m-1) ;

%p if type(m,'odd') and n = sm+1 then

%p return m-1 ;

%p else

%p return m;

%p end if;

%p end if;

%p end do:

%p end proc:

%p seq(A182021(n),n=3..80) ; # _R. J. Mathar_, Jul 12 2013

%t A093353[n_] := If[n < 1, 0, (n+Mod[n, 2])*(n+1)/2];

%t 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 Table[a[n], {n, 3, 80}] (* _Jean-François Alcover_, Apr 15 2023, after _R. J. Mathar_ *)

%K nonn,easy

%O 3,1

%A _N. J. A. Sloane_, Apr 06 2012