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Number of runs of 0's of odd length in all ternary words of length n.
2

%I #11 Feb 18 2016 10:40:00

%S 0,1,4,17,64,233,820,2825,9568,31985,105796,346913,1129312,3653657,

%T 11758132,37665881,120172096,382039649,1210689028,3825777329,

%U 12058462720,37918780361,118986517684,372650082857,1165021837984

%N Number of runs of 0's of odd length in all ternary words of length n.

%C a(n)=Sum(k*A119914(n,k),k>=0).

%C Binomial transform of A179608. - _Johannes W. Meijer_, Aug 01 2010

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-9).

%F G.f. = z(1-z)/[(1+z)(1-3z)^2].

%F a(n) = ((-1)^(n-1)+(3+4*n)*3^(n-1))/8. - _Johannes W. Meijer_, Aug 01 2010

%e a(2)=4 because in the nine ternary words of length 2, namely, 00, (0)1, (0)2, 1(0), 11, 12, 2(0), 21, 22, we have altogether 4 runs of 0's of odd length (shown between parentheses).

%p g:=z*(1-z)/(1-3*z)^2/(1+z): gser:=series(g,z=0,35): seq(coeff(gser,z,n),n=0..28);

%t LinearRecurrence[{5,-3,-9},{0,1,4},30] (* _Harvey P. Dale_, Feb 18 2016 *)

%Y Cf. A119914.

%K nonn,easy

%O 0,3

%A _Emeric Deutsch_, May 29 2006