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A105423 Number of compositions of n+2 having exactly two parts equal to 1. 8

%I

%S 1,0,3,3,9,15,31,57,108,199,366,666,1205,2166,3873,6891,12207,21537,

%T 37859,66327,115842,201743,350412,607140,1049545,1810428,3116655,

%U 5355219,9185349,15728547,26890375,45904773,78253896,133221079

%N Number of compositions of n+2 having exactly two parts equal to 1.

%C Column 2 of A105422.

%H J. J. Madden, <a href="http://arxiv.org/abs/1707.04351">A generating function for the distribution of runs in binary words</a>, arXiv:1707.04351 [math.CO], 2017. Theorem 1.1, r=1, k=2.

%F G.f.: (1-z)^3/(1-z-z^2)^3.

%F a(n) = (1/50) [(5n^2+21n+25)*Lucas(n) - (11n^2+30n+10)*Fibonacci(n) ]. - _Ralf Stephan_, Jun 01 2007

%e a(4)=9 because we have (1,1,4),(1,4,1),(4,1,1),(1,1,2,2),(1,2,1,2),(1,2,2,1),(2,1,1,2),(2,1,2,1) and (2,2,1,1).

%p G:=(1-z)^3/(1-z-z^2)^3: Gser:=series(G,z=0,42): 1,seq(coeff(Gser,z^n),n=1..40);

%t LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 0, 3, 3, 9, 15}, 40] (* _Jean-Fran├žois Alcover_, Jul 23 2018 *)

%Y Cf. A105422.

%K nonn,easy

%O 0,3

%A _Emeric Deutsch_, Apr 07 2005

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Last modified May 27 03:13 EDT 2022. Contains 354093 sequences. (Running on oeis4.)