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

2,23

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

Table of n, a(n) for n=2..120.

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.

Formula

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

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 *)

Crossrefs

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

Sequence in context: A271024 A125208 A333351 * A319993 A122776 A261609

Adjacent sequences: A193137 A193138 A193139 * A193141 A193142 A193143

Keyword

nonn

Author

N. J. A. Sloane, Jul 16 2011

Extensions

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

Status

approved