A051920 a(n) = binomial(n, floor(n/2)) + 1.

2, 2, 3, 4, 7, 11, 21, 36, 71, 127, 253, 463, 925, 1717, 3433, 6436, 12871, 24311, 48621, 92379, 184757, 352717, 705433, 1352079, 2704157, 5200301, 10400601, 20058301, 40116601, 77558761, 155117521, 300540196, 601080391, 1166803111

Offset: 0

Author

N. J. A. Sloane, Dec 18 1999

Keywords

• nonn

Comments

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

Links

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

Crossrefs

Cf. A001405, A006481, A007318, A193242.

Programs

Maple

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

Mathematica

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

Formula

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