A047859 a(n) = T(2, n), where T is the array given by A047858.

1, 4, 11, 27, 63, 143, 319, 703, 1535, 3327, 7167, 15359, 32767, 69631, 147455, 311295, 655359, 1376255, 2883583, 6029311, 12582911, 26214399, 54525951, 113246207, 234881023, 486539263, 1006632959, 2080374783, 4294967295, 8858370047, 18253611007, 37580963839

Offset

0,2

Comments

n-th difference of a(n), a(n-1), ..., a(0) is (3, 4, 5, ...).

From Gus Wiseman, Oct 14 2022: (Start)

Also the number of compositions of 2*(n+1) whose maximum part is n+1. These are compositions of 2*(n+1) whose maximum part equals the sum of their remaining parts. For example, the a(0) = 1 through a(2) = 11 compositions are:

(1,1) (2,2) (3,3)

(1,1,2) (1,2,3)

(1,2,1) (1,3,2)

(2,1,1) (2,1,3)

(2,3,1)

(3,1,2)

(3,2,1)

(1,1,1,3)

(1,1,3,1)

(1,3,1,1)

(3,1,1,1)

For length instead of maximum we have A001700.

These compositions are ranked by A357708. (End)

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..3000

Index entries for linear recurrences with constant coefficients, signature (5,-8,4).

Formula

Main diagonal of the array defined by: T(0, j) = j + 1 for j >= 0, T(i, 0) = i + 1 for i >= 0, T(i, j)= T(i-1, j-1) + T(i-1, j) + 1. a(n) = (n + 4)*2^(n-1) - 1. - Benoit Cloitre, Jun 17 2003

a(0) = 1, a(1) = 4, a(2) = 11, a(n) = 5*a(n-1) - 8*a(n-2) + 4*a(n-3). - Vincenzo Librandi, Sep 28 2011

G.f.: (1 - x - x^2)/((1 - x)*(1 - 2*x)^2). - Colin Barker, Aug 24 2016

a(n) = A045623(n) - 1. - Gus Wiseman, Oct 14 2022

E.g.f.: exp(x)*(exp(x)*(2 + x) - 1). - Stefano Spezia, Jan 02 2023

Prog

(Magma) [(n+4)*2^(n-1)-1: n in [0..30]]; // Vincenzo Librandi, Sep 28 2011
(PARI) Vec((1-x-x^2)/((1-x)*(1-2*x)^2) + O(x^40)) \\ Colin Barker, Aug 24 2016

Crossrefs

A011782 counts compositions.

Cf. A000120, A001511, A001700, A029931, A045623, A047858, A056239, A070939, A357708.

Sequence in context: A036890 A000253 A276691 * A100335 A340228 A080869

Adjacent sequences: A047856 A047857 A047858 * A047860 A047861 A047862

Keyword

nonn,easy

Author

Clark Kimberling

Status

approved