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Number of ternary words of length n and having exactly one run of 0's of odd length.
3

%I #12 Jan 15 2020 00:15:35

%S 0,1,4,13,40,117,332,921,2512,6761,18004,47525,124536,324317,840092,

%T 2166065,5562272,14232273,36300196,92321085,234192584,592695109,

%U 1496810732,3772761289,9492450672,23844342073,59804611060,149787196117

%N Number of ternary words of length n and having exactly one run of 0's of odd length.

%C Column 1 of A119914.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2,-4,-1).

%F a(n) = [z^n] z*(1 - z^2)/(1 - 2*z - z^2)^2.

%F a(n) = A006645(n+1) - A006645(n-1). - _R. J. Mathar_, Aug 07 2015

%F From _Peter Luschny_, Jan 14 2020: (Start)

%F a(n) = Sum_{k=0..n} A193737(n, k)*k.

%F Let h(k) = (1 + k)*exp((1 + k)*x)*(1 + x - 1/k)/4 then

%F a(n) = n!*[x^n](h(sqrt(2)) + h(-sqrt(2))). (End)

%e a(3) = 13 because we have 000, 011, 012, 021, 022, 101, 102, 110, 120, 201, 202, 210 and 220 (for example, 001, 020 do not qualify).

%p g := z*(1-z^2)/(1-2*z-z^2)^2:

%p gser := series(g,z=0,34):

%p seq(coeff(gser,z,n), n=0..30);

%t LinearRecurrence[ {4, -2, -4, -1}, {0, 1, 4, 13}, 28] (* _Peter Luschny_, Jan 14 2020 *)

%Y Cf. A119914, A193737, A006645.

%K nonn,easy

%O 0,3

%A _Emeric Deutsch_, May 29 2006