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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved