A153893 a(n) = 3*2^n - 1.

2, 5, 11, 23, 47, 95, 191, 383, 767, 1535, 3071, 6143, 12287, 24575, 49151, 98303, 196607, 393215, 786431, 1572863, 3145727, 6291455, 12582911, 25165823, 50331647, 100663295, 201326591, 402653183, 805306367, 1610612735, 3221225471

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Jan 03 2009

Keywords

• nonn
• easy

Comments

A020944(a(n)) = 0. - Reinhard Zumkeller, Mar 13 2011

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

Number of distinct continued fractions of n terms chosen from {1,2}. - Clark Kimberling, Jul 20 2015

Also, the decimal representation of the x-axis, from the origin to the right edge, of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 643", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. See A283508. - Robert Price, Mar 09 2017

This sequence has been used by the ninth-century mathematician Thabit ibn Qurra to devise the first method to construct amicable pairs (see Tattersall). - Stefano Spezia, Jul 18 2025

References

• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 138.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• 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. 2-5, 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. A283508.

Programs

Magma

[3*2^n-1: n in [0..30]]; // Vincenzo Librandi, Oct 28 2011

Mathematica

Table[3*2^n - 1 , {n,0,25}] (* G. C. Greubel, Sep 01 2016 *)
LinearRecurrence[{3,-2},{2,5},40] (* Harvey P. Dale, Mar 01 2024 *)

PARI

a(n)=3*2^n-1 \\ Charles R Greathouse IV, Sep 24 2015

Formula

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

a(n) = A083329(n+1).

a(n) = A055010(n+1).

G.f.: (2 - x)/((1-x)(1-2x)). - R. J. Mathar, Feb 13 2009

a(n) = A083416(2n) = A033484(n) + 1. - Philippe Deléham, Apr 14 2013

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

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

E.g.f.: 3*exp(2*x) - exp(x). (End)

Extensions

Edited by N. J. A. Sloane, Feb 14 2009