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 A105423 Number of compositions of n+2 having exactly two parts equal to 1. 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Column 2 of A105422. LINKS Table of n, a(n) for n=0..33. 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). 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 EXAMPLE 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). 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); MATHEMATICA LinearRecurrence[{3, 0, -5, 0, 3, 1}, {1, 0, 3, 3, 9, 15}, 40] (* Jean-François Alcover, Jul 23 2018 *) CROSSREFS Cf. A105422. Sequence in context: A264098 A223209 A233026 * A147471 A062510 A000200 Adjacent sequences: A105420 A105421 A105422 * A105424 A105425 A105426 KEYWORD nonn,easy AUTHOR Emeric Deutsch, Apr 07 2005 STATUS approved

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Last modified May 22 06:24 EDT 2024. Contains 372743 sequences. (Running on oeis4.)