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A193140 Number of isonemal satins of exact period n.

%I A193140 #33 Mar 21 2021 13:08:54
%S A193140 0,0,0,1,0,0,1,0,1,0,1,1,0,1,1,1,0,0,1,1,0,0,3,1,1,0,1,1,1,0,1,1,1,1,
%T A193140 1,1,0,1,3,1,1,0,1,1,0,0,3,0,1,1,1,1,0,1,3,1,1,0,3,1,0,1,1,3,1,0,1,1,
%U A193140 1,0,3,1,1,1,1,1,1,0,3,0,1,0,3,3,0,1,3,1,1,1,1,1,0,1,3,1,0,1,1,1,1,0,3,3,1,0,1,1,1,1,3,1,1,1,1,1,0,1,7
%N A193140 Number of isonemal satins of exact period n.
%C A193140 On page 153 of Grünbaum and Shephard (1980) is Table 3 which is a list of all the (n,s)-satins with n<=100. - _Michael Somos_, Dec 05 2014
%D A193140 B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-98 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
%H A193140 B. Grünbaum and G. C. Shephard, <a href="http://www.jstor.org/stable/2690105">Satins and twills: an introduction to the geometry of fabrics</a>, Math. Mag., 53 (1980), 139-161. See Theorem 5, page 152.
%F A193140 a(n) = A086669(n) - 1. - _Andrey Zabolotskiy_, Dec 25 2018
%p A193140 #A193138
%p A193140 U:=proc(n) local j,p3,i,t1,t2,al,even;
%p A193140 t1:=ifactors(n)[2];
%p A193140 t2:=nops(t1);
%p A193140 if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
%p A193140 j:=t2-even;
%p A193140 p3:=0;
%p A193140 for i from 1 to t2 do if t1[i][1] mod 4 = 3 then p3:=1; fi; od:
%p A193140 if (al >= 2) or (p3=1) then RETURN(0) else RETURN(2^(j-1)); fi;
%p A193140 end;
%p A193140 #A193139:
%p A193140 V:=proc(n) local j,i,t1,t2,al,even;
%p A193140 t1:=ifactors(n)[2];
%p A193140 t2:=nops(t1);
%p A193140 if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
%p A193140 j:=t2-even;
%p A193140 if (al <= 1) then RETURN(2^(j-1)-1); fi;
%p A193140 if (al = 2) then RETURN(2^j-1); fi;
%p A193140 if (al >= 3) then RETURN(2^(j+1)-1); fi;
%p A193140 end;
%p A193140 #A193140:
%p A193140 [seq(U(n)+V(n), n=3..120)];
%t A193140 a[n_] := 2^With[{f = FactorInteger[n]}, Length@f - If[
%t A193140   f[[1, 1]] == 2 && f[[1, 2]] > 1,
%t A193140   Boole[f[[1, 2]] == 2],
%t A193140   Boole[f[[1, 1]] == 2] + Boole[AnyTrue[f[[;; , 1]], Mod[#, 4] == 3 &]]
%t A193140 ]] - 1;
%t A193140 Table[a[n], {n, 2, 100}]
%t A193140 (* _Andrey Zabolotskiy_, Mar 21 2021 *)
%Y A193140 Cf. A193138, A193139, A086669, A000046, A005441, A262589.
%K A193140 nonn
%O A193140 2,23
%A A193140 _N. J. A. Sloane_, Jul 16 2011
%E A193140 a(2) = 0 prepended and name edited by _Andrey Zabolotskiy_, Mar 21 2021