A213332 Number of isomorphism classes of reduced Witt rings of fields with n orderings.

1, 1, 1, 2, 2, 3, 3, 6, 6, 9, 9, 16, 16, 24, 24, 42, 42, 64, 64, 105, 105, 159, 159, 258, 258, 390, 390, 614, 614, 925, 925, 1441, 1441, 2162, 2162, 3317, 3317, 4951, 4951, 7526, 7526, 11191, 11191, 16841, 16841, 24923, 24923, 37253, 37253, 54912, 54912, 81493, 81493, 119629, 119629, 176549

Offset

1,4

Comments

The number with 2m+1 orderings is the same as the number with 2m orderings (cf. A213331).

Links

Table of n, a(n) for n=1..56.

Thomas C. Craven, An application of Pólya's theory of counting to an enumeration problem arising in quadratic form theory, J. Combin. Theory Ser. A 29 (1980), no. 2, 174--181. MR0583956 (81j:10027).

Maple

read transforms;
w:=proc(n) option remember; global did; local v; # did(n, d)=1 if d|n otherwise 0
if n=1 then 1 elif (n mod 2) = 1 then w(n-1);
else v:=n/2;
(1/n)* ( add(2*i*w(i)*did(v, i), i=1..v) +
add( add(2*i*w(i)*w(n-2*k)*did(k, i), i=1..k), k=1..v-1));
fi; end;
[seq(w(n), n=1..100)];

Crossrefs

Cf. A213331.

Sequence in context: A038716 A168659 A035642 * A133392 A101199 A032155

Adjacent sequences: A213329 A213330 A213331 * A213333 A213334 A213335

Keyword

nonn

Author

N. J. A. Sloane, Jun 09 2012

Status

approved