A083705 a(n) = 2*a(n-1) - 1 with a(0) = 10.

10, 19, 37, 73, 145, 289, 577, 1153, 2305, 4609, 9217, 18433, 36865, 73729, 147457, 294913, 589825, 1179649, 2359297, 4718593, 9437185, 18874369, 37748737, 75497473, 150994945, 301989889, 603979777, 1207959553, 2415919105, 4831838209, 9663676417, 19327352833

Offset: 0

Author

N. J. A. Sloane, Jun 15 2003

Keywords

• nonn
• easy

Comments

An Engel expansion of 2/9 to the base 2 as defined in A181565, with the associated series expansion 2/9 = 2/10 + 2^2/(10*19) + 2^3/(10*19*37) + 2^4/(10*19*37*73) + ... . - Peter Bala, Oct 29 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

A020737, A083575, A083683, A083686, A168596, A181565, A195744.

Programs

Magma

[9*2^n+1 : n in [0..30]]; // Vincenzo Librandi, Nov 03 2011

Mathematica

s=10;lst={s};Do[s=s+(s-1);AppendTo[lst,s],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 30 2009 *)
NestList[2#-1&,10,40]  (* Harvey P. Dale, Mar 13 2011 *)

Python

from itertools import accumulate
def f(an, _): return 2*an - 1
print(list(accumulate([10]*32, f))) # Michael S. Branicky, Oct 19 2021

Formula

From R. J. Mathar, Aug 01 2009: (Start)

a(n) = 1 + 9*2^n = 3*a(n-1) - 2*a(n-2).

G.f.: -(-10+11*x)/((2*x-1)*(x-1)). (End)

E.g.f.: exp(x)*(1 + 9*exp(x)). - Stefano Spezia, Oct 08 2022