0,1

With the exception of the initial 2s, these are numbers such that if Pascal's triangle is written in base a(n) - 1, the first n - 2 rows give the digits of the powers of a(n) written in that base. This is most often noticed for the powers of 11 since of course we use decimal. - Alonso del Arte, Jul 10 2011

Alois P. Heinz, Table of n, a(n) for n = 0..1000

G.f.: -((2*x-1)*(3*x-1) +(x-1)*sqrt(1 - 4*x^2))/(2*x*(x-1)*(2*x-1)). - Thomas Baruchel, Jun 26 2018

0 = 1 +a(n)*(-2 +4*a(n+1) -2*a(n+2)) +a(n+1)*(-1 -2*a(n+1) +a(n+2)) +a(n+2) for all n>=0. - Michael Somos, Jun 30 2018

a:= proc(n) option remember; `if`(n<3, [2, 2, 3][n+1],

((n^2+n-4)*a(n-1) +2*(n-1)*(2*n-5)*a(n-2)

-4*(n-1)*(n-2)*a(n-3)) / ((n+1)*(n-2)))

end:

seq(a(n), n=0..50); # Alois P. Heinz, Mar 03 2014

a[n_] := a[n] = (4(n-1) a[n-2] + 2a[n-1] - 3n + 3)/(n+1); a[0] = a[1] = 2; Array[a, 50, 0] (* Jean-François Alcover, Jan 19 2017 *)

Table[Binomial[n, Floor[n/2]], {n, 0, 40}]+1 (* Harvey P. Dale, Jan 20 2019 *)

(PARI) a(n)=binomial(n, n\2)+1 \\ Charles R Greathouse IV, Feb 05 2013

Cf. A001405, A006481, A007318, A193242.

Sequence in context: A243856 A173433 A053638 * A286350 A023105 A281723

Adjacent sequences: A051917 A051918 A051919 * A051921 A051922 A051923

N. J. A. Sloane, Dec 18 1999

approved