A340228 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.

4, 11, 27, 64, 149, 342, 775, 1736, 3849, 8458, 18443, 39948, 86029, 184334, 393231, 835600, 1769489, 3735570, 7864339, 16515092, 34603029, 72351766, 150994967, 314572824, 654311449, 1358954522, 2818572315, 5838471196, 12079595549, 24964497438, 51539607583, 106300440608

Offset

1,1

Links

Table of n, a(n) for n=1..32.

Index entries for linear recurrences with constant coefficients, signature (6,-13,12,-4).

Index entries for sequences related to compositions.

Formula

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

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

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

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

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

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

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

Example

Illustrations for n = 1..4:
      _           _ _
     |_|         |_ _|
                 |_|_|
  a(1) = 4     a(2) = 11
    _ _ _       _ _ _ _
   |_ _ _|     |_ _ _ _|
   |_ _|_|     |_ _ _|_|
   |_|_ _|     |_|_ _ _|
   |_|_|_|     |_ _|_ _|
               |_ _|_|_|
               |_|_ _|_|
               |_|_|_ _|
               |_|_|_|_|
  a(3) = 27    a(4) = 64

Mathematica

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

Crossrefs

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

Cf. A338969.

Sequence in context: A276691 A047859 A100335 * A080869 A137229 A301874

Adjacent sequences: A340225 A340226 A340227 * A340229 A340230 A340231

Keyword

nonn,easy

Author

Stefano Spezia, Jan 01 2021

Status

approved