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

Author

Clark Kimberling

Keywords

• nonn
• easy

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).

Crossrefs

A011782 counts compositions.

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

Programs

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

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