A153894 a(n) = 5*2^n - 1.

4, 9, 19, 39, 79, 159, 319, 639, 1279, 2559, 5119, 10239, 20479, 40959, 81919, 163839, 327679, 655359, 1310719, 2621439, 5242879, 10485759, 20971519, 41943039, 83886079, 167772159, 335544319, 671088639, 1342177279, 2684354559

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

• nonn
• easy

Comments

a(n) is the total number of symbols required in the fully-expanded von Neumann definition of ordinal n + 1, where the string "{}" is used to represent the empty set and spaces are ignored. - Ely Golden, Nov 14 2019

a(n) converted to binary is 100 followed by n ones. - Alexandre Herrera, Oct 06 2023

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Bernard Monjardet, Acyclic domains of linear orders: a survey, in "The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn", edited by Steven Brams, William V. Gehrlein and Fred S. Roberts, Springer, 2009, pp. 139-160. This version: . - _N. J. A. Sloane_, Feb 07 2009
• Gennady Eremin, Partitioning the set of natural numbers into Mersenne trees and into arithmetic progressions; Natural Matrix and Linnik's constant, arXiv:2405.16143 [math.CO], 2024. See pp. 3-5, 14.
• Gennady Eremin, Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers, arXiv:2501.17090 [math.GM], 2025. See pp. 3, 11.
• Index entries for linear recurrences with constant coefficients, signature (3,-2).

Crossrefs

Cf. A052549, A309779.

Programs

Magma

[5*2^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 28 2011

Mathematica

a=4;lst={a};Do[a=a*2+1;AppendTo[lst,a],{n,5!}];lst
LinearRecurrence[{3,-2},{4,9}, 25] (* or *) Table[5*2^n - 1, {n,0,25}] (* G. C. Greubel, Sep 01 2016 *)

PARI

a(n)=5*2^n-1 \\ Charles R Greathouse IV, Oct 07 2015

Formula

a(n) = 2*a(n-1) + 1, n>0.

a(n) = A052549(n+1).

G.f.: (4 - 3*x) / ( (2*x-1)*(x-1) ). - R. J. Mathar, Oct 22 2011

a(n) + a(n-1)^2 = A309779(n), a perfect square. - Vincenzo Librandi, Oct 28 2011

From G. C. Greubel, Sep 01 2016: (Start)

a(n) = 3*a(n-1) - 2*a(n-2).

E.g.f.: 5*exp(2*x) - exp(x). (End)

Extensions

Edited by N. J. A. Sloane, Feb 07 2009

Definition corrected by Franklin T. Adams-Watters, Apr 22 2009