A340228 - OEIS

%I #8 Jan 20 2021 18:50:33

%S 4,11,27,64,149,342,775,1736,3849,8458,18443,39948,86029,184334,

%T 393231,835600,1769489,3735570,7864339,16515092,34603029,72351766,

%U 150994967,314572824,654311449,1358954522,2818572315,5838471196,12079595549,24964497438,51539607583,106300440608

%N a(n) is the sum of the lengths of all the segments used to draw a rectangle of height 2^(n-1) and width n divided into 2^(n-1) rectangles of unit height, in turn, divided into rectangles of unit height and lengths corresponding to the parts of the compositions of n.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (6,-13,12,-4).

%H <a href="/index/Com#compositions">Index entries for sequences related to compositions</a>.

%F O.g.f.: x*(4 - 13*x + 13*x^2 - 3*x^3)/(1 - 3*x + 2*x^2)^2.

%F E.g.f.: (exp(2*x)*(3 + 6*x) + 4*x*exp(x) - 3)/4.

%F a(n) = 6*a(n-1) - 13*a(n-2) + 12*a(n-3) - 4*a(n-4) for n > 4.

%F a(n) = n + 3*(n + 1)*2^(n-2).

%F a(n) = A001792(n) + A188626(n).

%F a(n) = A045623(n) + A215149(n).

%F a(n) = A006127(n) + A053220(n).

%e Illustrations for n = 1..4:

%e       _           _ _

%e      |_|         |_ _|

%e                  |_|_|

%e   a(1) = 4     a(2) = 11

%e     _ _ _       _ _ _ _

%e    |_ _ _|     |_ _ _ _|

%e    |_ _|_|     |_ _ _|_|

%e    |_|_ _|     |_|_ _ _|

%e    |_|_|_|     |_ _|_ _|

%e                |_ _|_|_|

%e                |_|_ _|_|

%e                |_|_|_ _|

%e                |_|_|_|_|

%e   a(3) = 27    a(4) = 64

%t LinearRecurrence[{6,-13,12,-4},{4,11,27,64},32]

%Y Cf. A000079, A001792, A006127, A011782, A045623, A053220, A188626, A215149, A228525.

%Y Cf. A338969.

%K nonn,easy

%O 1,1

%A _Stefano Spezia_, Jan 01 2021