A193140 Number of isonemal satins of exact period n.

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, 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, 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

Offset: 2

Author

N. J. A. Sloane, Jul 16 2011

Keywords

• nonn

Comments

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

References

• 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.

Links

• B. Grünbaum and G. C. Shephard, Satins and twills: an introduction to the geometry of fabrics, Math. Mag., 53 (1980), 139-161. See Theorem 5, page 152.

Crossrefs

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

Programs

Maple

#A193138
U:=proc(n) local j,p3,i,t1,t2,al,even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
j:=t2-even;
p3:=0;
for i from 1 to t2 do if t1[i][1] mod 4 = 3 then p3:=1; fi; od:
if (al >= 2) or (p3=1) then RETURN(0) else RETURN(2^(j-1)); fi;
end;
#A193139:
V:=proc(n) local j,i,t1,t2,al,even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
j:=t2-even;
if (al <= 1) then RETURN(2^(j-1)-1); fi;
if (al = 2) then RETURN(2^j-1); fi;
if (al >= 3) then RETURN(2^(j+1)-1); fi;
end;
#A193140:
[seq(U(n)+V(n), n=3..120)];

Mathematica

a[n_] := 2^With[{f = FactorInteger[n]}, Length@f - If[
  f[[1, 1]] == 2 && f[[1, 2]] > 1,
  Boole[f[[1, 2]] == 2],
  Boole[f[[1, 1]] == 2] + Boole[AnyTrue[f[[;; , 1]], Mod[#, 4] == 3 &]]
]] - 1;
Table[a[n], {n, 2, 100}]
(* Andrey Zabolotskiy, Mar 21 2021 *)

Formula

a(n) = A086669(n) - 1. - Andrey Zabolotskiy, Dec 25 2018

Extensions

a(2) = 0 prepended and name edited by Andrey Zabolotskiy, Mar 21 2021