A073996 Number of strings of length n over GF(4) with trace 0 and subtrace 1.

0, 1, 3, 12, 60, 256, 1008, 4032, 16320, 65536, 261888, 1047552, 4193280, 16777216, 67104768, 268419072, 1073725440, 4294967296, 17179803648, 68719214592, 274877644800, 1099511627776, 4398045462528, 17592181850112, 70368739983360, 281474976710656, 1125899890065408, 4503599560261632, 18014398442373120

Offset

1,3

Comments

Same as the number of strings of length n over GF(4) with trace 0 and subtrace x where x=RootOf(z^2+z+1). Same as the number of strings of length n over GF(4) with trace 0 and subtrace y where y=1+x.

Links

Table of n, a(n) for n=1..29.

F. Ruskey Strings over GF(4) with given trace and subtrace

Formula

a(n; t, s) = a(n-1; t, s) + a(n-1; t-1, s-(t-1)) + a(n-1; t-2, s-2(t-2)) + a(n-1; t-3, s-3(t-3)) where t is the trace and s is the subtrace. Note that all operations involving operands t or s are carried out over GF(4).

G.f.: (6*q^2-3*q+1)*q^2/[(1-2q)(1-4q)(1+4q^2)]. - Lawrence Sze, Oct 24 2004

Example

a(2;0,1)=1 since the one 4-ary string of trace 0, subtrace 1 and length 2 is { 11 }.

Mathematica

CoefficientList[Series[x^2 (6x^2-3x+1)/((1-2x)(1-4x)(1+4x^2)), {x, 0, 30}], x]  (* Harvey P. Dale, Apr 03 2011 *)

Crossrefs

Cf. A073995, A073997, A073998, A073999, A074000.

Sequence in context: A127918 A069944 A253171 * A003483 A278395 A128602

Adjacent sequences: A073993 A073994 A073995 * A073997 A073998 A073999

Keyword

easy,nonn

Author

Frank Ruskey and Nate Kube, Aug 16 2002

Extensions

More terms from Harvey P. Dale, Apr 03 2011

More terms from Max Alekseyev, Apr 16 2013

Status

approved