A105423 Number of compositions of n+2 having exactly two parts equal to 1.
1, 0, 3, 3, 9, 15, 31, 57, 108, 199, 366, 666, 1205, 2166, 3873, 6891, 12207, 21537, 37859, 66327, 115842, 201743, 350412, 607140, 1049545, 1810428, 3116655, 5355219, 9185349, 15728547, 26890375, 45904773, 78253896, 133221079
Offset: 0
Examples
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).
Links
- Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 11.
- J. J. Madden, A generating function for the distribution of runs in binary words, arXiv:1707.04351 [math.CO], 2017. Theorem 1.1, r=1, k=2.
- Index entries for linear recurrences with constant coefficients, signature (3, 0, -5, 0, 3, 1).
Crossrefs
Cf. A105422.
Programs
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Maple
G:=(1-z)^3/(1-z-z^2)^3: Gser:=series(G,z=0,42): 1,seq(coeff(Gser,z^n),n=1..40);
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Mathematica
LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 0, 3, 3, 9, 15}, 40] (* Jean-François Alcover, Jul 23 2018 *)
Formula
G.f.: (1-z)^3/(1-z-z^2)^3.
a(n) = (1/50) [(5n^2+21n+25)*Lucas(n) - (11n^2+30n+10)*Fibonacci(n) ]. - Ralf Stephan, Jun 01 2007
Comments