login
a(n) = 2^(n+2)*(2^(n+1)-1).
8

%I #64 Jul 26 2024 21:16:31

%S 4,24,112,480,1984,8064,32512,130560,523264,2095104,8384512,33546240,

%T 134201344,536838144,2147418112,8589803520,34359476224,137438429184,

%U 549754765312,2199021158400,8796088827904,35184363700224,140737471578112,562949919866880

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

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

%C 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

%C 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

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

%H Harry J. Smith, <a href="/A059153/b059153.txt">Table of n, a(n) for n = 0..200</a>

%H J. Wallgren, <a href="/A059126/a059126.txt">Hierarchical sequences</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ElementaryCellularAutomaton.html">Elementary Cellular Automaton</a>

%H S. Wolfram, <a href="http://wolframscience.com/">A New Kind of Science</a>

%H <a href="/index/Ce#cell">Index entries for sequences related to cellular automata</a>

%H <a href="https://oeis.org/wiki/Index_to_2D_5-Neighbor_Cellular_Automata">Index to 2D 5-Neighbor Cellular Automata</a>

%H <a href="https://oeis.org/wiki/Index_to_Elementary_Cellular_Automata">Index to Elementary Cellular Automata</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,-8).

%F a(n) = A173787(2*n+3,n+2) = 4*A006516(n+1). - _Reinhard Zumkeller_, Feb 28 2010

%F From _Colin Barker_, Apr 28 2013: (Start)

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

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

%F a(n) = 2*A020522(n+1). - _Hussam al-Homsi_, Jun 06 2021

%F E.g.f.: 4*exp(2*x)*(2*exp(2*x) - 1). - _Elmo R. Oliveira_, Dec 10 2023

%t 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 *)

%o (PARI) { for (n = 0, 200, write("b059153.txt", n, " ", 2^(n + 2)*(2^(n + 1) - 1)); ) } \\ _Harry J. Smith_, Jun 25 2009

%Y Cf. A006516, A020522, Equals A045687(2n+4), A059126, A163663, A173787, A272702.

%K easy,nonn

%O 0,1

%A _Jonas Wallgren_, Feb 02 2001

%E Revised by _Henry Bottomley_, Jun 27 2005