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Number of strings over Z_3 of length n with trace 1 and subtrace 1.
5

%I #18 May 03 2019 03:04:17

%S 0,1,3,6,21,81,252,729,2187,6642,19845,59049,176904,531441,1594323,

%T 4780782,14344533,43046721,129146724,387420489,1162261467,3486843450,

%U 10460471301,31381059609,94143001680,282429536481,847288609443,2541864234006,7625594296341

%N Number of strings over Z_3 of length n with trace 1 and subtrace 1.

%C Same as number of strings over Z_3 of length n with trace 2 and subtrace 1. Same as number of strings over GF(3) of length n with trace 1 and subtrace 1. Same as number of strings over GF(3) of length n with trace 2 and subtrace 1.

%H F. Ruskey, <a href="http://combos.org/TSstringZ3">Strings over Z_3 with given trace and subtrace</a>

%H F. Ruskey, <a href="http://combos.org/TSstringF3">Strings over GF(3) with given trace and subtrace</a>

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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,27,-36,27).

%F a(n; t, s) = a(n-1; t, s) + a(n-1; t+2, s+2t+1) + a(n-1; t+1, s+t+1) where t is the trace and s is the subtrace.

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

%e a(2;2,1)=1 since the one ternary string of trace 2, subtrace 1 and length 2 is { 11 }.

%Y Cf. A073947, A073948, A073949, A073950, A073952.

%K easy,nonn

%O 1,3

%A _Frank Ruskey_ and Nate Kube, Aug 15 2002

%E Terms a(21) onward from _Max Alekseyev_, Apr 09 2013