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Number of strings of length n over Z_5 with trace 1 and subtrace 2.
7

%I #22 May 03 2019 07:17:35

%S 0,0,1,20,125,650,3150,15625,78000,390625,1952500,9762500,48815625,

%T 244125000,1220703125,6103593750,30517656250,152587890625,

%U 762939062500,3814697265625,19073484375000,95367421875000,476837119140625,2384185742187500,11920928955078125

%N Number of strings of length n over Z_5 with trace 1 and subtrace 2.

%C Same as the number of strings of length n over Z_5 with: trace 2 and subtrace 3, trace 3 and subtrace 3, or trace 4 and subtrace 2.

%C Same as the number of strings of length n over GF(5) with: trace 1 and subtrace 2, trace 2 and subtrace 3, trace 3 and subtrace 3, or trace 4 and subtrace 2.

%H Max Alekseyev, <a href="http://home.gwu.edu/~maxal/gpscripts/">PARI/GP scripts for miscellaneous math problems</a>

%H F. Ruskey, <a href="http://combos.org/TSstringZ5">Strings over Z_5 with given trace and subtrace</a>

%H F. Ruskey, <a href="http://combos.org/TSstringF5">Strings over GF(5) with given trace and subtrace</a>

%F a(n; t, s) = a(n-1; t, s) + a(n-1; t+4, s+4t+1) + a(n-1; t+3, s+3t+4) + a(n-1; t+2, s+2t+4) + a(n-1; t+1, s+t+1).

%F Empirical g.f.: -x^3*(25*x^4-50*x^3+40*x^2-10*x-1) / ((5*x-1)*(5*x^2-1)*(25*x^4-25*x^3+15*x^2-5*x+1)). - _Colin Barker_, Nov 26 2014

%e a(3;1,2)=1 since the one 5-ary string of trace 1, subtrace 2 and length 3 is { 222 }.

%Y Cf. A073963, A073964, A073965, A073966, A073967, A073969, A073970.

%K easy,nonn

%O 1,4

%A _Frank Ruskey_ and Nate Kube, Aug 15 2002

%E Terms a(11) onward from _Max Alekseyev_, Apr 09 2013