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

Author

Frank Ruskey and Nate Kube, Aug 16 2002

Keywords

• easy
• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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

Extensions

More terms from Harvey P. Dale, Apr 03 2011

More terms from Max Alekseyev, Apr 16 2013