A005441 -id:A005441 - cp's OEIS frontend

A362294 Erroneous version of A005441.

1, 1, 4, 4, 9, 8, 55, 21, 105, 62, 429, 196

Offset: 2

dead

Included in accordance with OEIS policy of including published but erroneous sequences to serve as pointers to the correct versions.

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-97 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. [the term a(10) = 105 given there is probably erroneous]

A193140 Number of isonemal satins of exact period n.

Original entry on oeis.org

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

N. J. A. Sloane, Jul 16 2011

nonn

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

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.

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.

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

#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)];

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 *)

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

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

A262589 Number of other isonemal fabrics of exact period n.

Original entry on oeis.org

0, 0, 2, 0, 4, 0, 40, 0, 64, 0, 316, 6, 448, 90, 3529, 14, 3836, 0

Offset: 2

N. J. A. Sloane, Oct 21 2015

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-97 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984. [gives different a(10) which is probably erroneous]

Branko Grünbaum and G. C. Shephard, A catalogue of isonemal fabrics, Discrete Geometry and Convexity, 440 (1985), 279-298.
B. Grünbaum and G. C. Shephard, An extension to the catalogue of isonemal fabrics, Discrete Mathematics, 60, (1986), 155-192.

Cf. A000046, A005441, A193140.

A005441(n) = A000046(n) + A193140(n) + a(n).

a(14)-a(19) using formula added, a(10) corrected by Andrey Zabolotskiy, Apr 14 2023

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A362294 Erroneous version of A005441.

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A193140 Number of isonemal satins of exact period n.

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A262589 Number of other isonemal fabrics of exact period n.

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Author

Keywords

References

Links

Crossrefs

Formula

Extensions