A059153 a(n) = 2^(n+2)*(2^(n+1)-1).

4, 24, 112, 480, 1984, 8064, 32512, 130560, 523264, 2095104, 8384512, 33546240, 134201344, 536838144, 2147418112, 8589803520, 34359476224, 137438429184, 549754765312, 2199021158400, 8796088827904, 35184363700224, 140737471578112, 562949919866880

Offset: 0

Author

Jonas Wallgren, Feb 02 2001

Keywords

• easy
• nonn

Comments

A hierarchical sequence (S(W'2{2}c) - see A059126).

a(n) written in base 2: 100, 11000, 1110000, ..., i.e., (n+1) times 1 and (n+2) times 0 (see A163663). - Jaroslav Krizek, Aug 12 2009

Also, the number of active (ON, black) cells at stage 2^n-1 of the two-dimensional cellular automaton defined by "Rule 513", based on the 5-celled von Neumann neighborhood. - Robert Price, May 04 2016

References

• S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

Links

• Harry J. Smith, Table of n, a(n) for n = 0..200
• J. Wallgren, Hierarchical sequences
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Index entries for sequences related to cellular automata
• Index to 2D 5-Neighbor Cellular Automata
• Index to Elementary Cellular Automata
• Index entries for linear recurrences with constant coefficients, signature (6,-8).

Crossrefs

Cf. A006516, A020522, A059126, A163663, A173787, A272702.

Programs

Mathematica

Table[2^(n + 2)*(2^(n + 1) - 1), {n, 0, 23}] (* and *) LinearRecurrence[{6, -8}, {4, 24}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)

PARI

a(n) = { 2^(n + 2)*(2^(n + 1) - 1) } \\ Harry J. Smith, Jun 25 2009

Formula

a(n) = A173787(2*n+3,n+2) = 4*A006516(n+1). - Reinhard Zumkeller, Feb 28 2010

From Colin Barker, Apr 28 2013: (Start)

a(n) = 6*a(n-1) - 8*a(n-2).

G.f.: 4 / ((2*x-1)*(4*x-1)). (End)

a(n) = 2*A020522(n+1). - Hussam al-Homsi, Jun 06 2021

E.g.f.: 4*exp(2*x)*(2*exp(2*x) - 1). - Elmo R. Oliveira, Dec 10 2023

Extensions

Revised by Henry Bottomley, Jun 27 2005