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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
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.
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
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.)