login
Number of isonemal satins of exact period n.
6

%I #33 Mar 21 2021 13:08:54

%S 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 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 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 Number of isonemal satins of exact period n.

%C 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 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 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 a(n) = A086669(n) - 1. - _Andrey Zabolotskiy_, Dec 25 2018

%p #A193138

%p U:=proc(n) local j,p3,i,t1,t2,al,even;

%p t1:=ifactors(n)[2];

%p t2:=nops(t1);

%p if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;

%p j:=t2-even;

%p p3:=0;

%p for i from 1 to t2 do if t1[i][1] mod 4 = 3 then p3:=1; fi; od:

%p if (al >= 2) or (p3=1) then RETURN(0) else RETURN(2^(j-1)); fi;

%p end;

%p #A193139:

%p V:=proc(n) local j,i,t1,t2,al,even;

%p t1:=ifactors(n)[2];

%p t2:=nops(t1);

%p if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;

%p j:=t2-even;

%p if (al <= 1) then RETURN(2^(j-1)-1); fi;

%p if (al = 2) then RETURN(2^j-1); fi;

%p if (al >= 3) then RETURN(2^(j+1)-1); fi;

%p end;

%p #A193140:

%p [seq(U(n)+V(n), n=3..120)];

%t a[n_] := 2^With[{f = FactorInteger[n]}, Length@f - If[

%t f[[1, 1]] == 2 && f[[1, 2]] > 1,

%t Boole[f[[1, 2]] == 2],

%t Boole[f[[1, 1]] == 2] + Boole[AnyTrue[f[[;; , 1]], Mod[#, 4] == 3 &]]

%t ]] - 1;

%t Table[a[n], {n, 2, 100}]

%t (* _Andrey Zabolotskiy_, Mar 21 2021 *)

%Y Cf. A193138, A193139, A086669, A000046, A005441, A262589.

%K nonn

%O 2,23

%A _N. J. A. Sloane_, Jul 16 2011

%E a(2) = 0 prepended and name edited by _Andrey Zabolotskiy_, Mar 21 2021