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

0,3

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