A073999 Number of strings of length n over GF(4) with trace 1 and subtrace x where x = RootOf(z^2+z+1).

0, 0, 3, 16, 60, 240, 1008, 4096, 16320, 65280, 261888, 1048576, 4193280, 16773120, 67104768, 268435456, 1073725440, 4294901760, 17179803648, 68719476736, 274877644800, 1099510579200, 4398045462528, 17592186044416, 70368739983360, 281474959933440, 1125899890065408, 4503599627370496, 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 x and subtrace 1. Same as the number of strings of length n over GF(4) with trace y and subtrace 1 where y = 1+x. Same as the number of strings of length n over GF(4) with trace 1 and subtrace y. Same as the number of strings of length n over GF(4) with trace x and subtrace x. Same as the number of strings of length n over GF(4) with trace y and subtrace y.

Links

• F. Ruskey, Strings over GF(4) with given trace and subtrace
• Index entries for linear recurrences with constant coefficients, signature (6,-12,24,-32).

Crossrefs

Cf. A073995, A073996, A073997, A073998.

Programs

Mathematica

LinearRecurrence[{6,-12,24,-32},{0,0,3,16},30] (* Harvey P. Dale, Mar 12 2019 *)

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

a(n) = -2^(n-3) +( (-2i)^n + (2i)^n +4^n )/16 with i=sqrt(-1). - R. J. Mathar, Nov 18 2011

Extensions

More terms from Max Alekseyev, Apr 16 2013