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A073996
Number of strings of length n over GF(4) with trace 0 and subtrace 1.
9
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.
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
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