A140253 a(2*n) = 2*(2*4^(n-1)-1) and a(2*n-1) = 2*4^(n-1)-1.

-1, 1, 2, 7, 14, 31, 62, 127, 254, 511, 1022, 2047, 4094, 8191, 16382, 32767, 65534, 131071, 262142, 524287, 1048574, 2097151, 4194302, 8388607, 16777214, 33554431, 67108862, 134217727, 268435454, 536870911

Offset: 0

Author

Paul Curtz, Jun 23 2008

Keywords

• sign
• easy

Comments

The inverse binomial transform is 1, 1, 4, -2, 10, -14, 34, -62 which leads to (-1)^(n+1)*A135440(n).

For n > 0: A266161(a(n)) = n and A266161(m) < n for m < a(n). - Reinhard Zumkeller, Dec 22 2015

Also, the decimal representation of the diagonal from the corner to the origin of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 673", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Jul 23 2017

References

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

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
• N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
• Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
• S. Wolfram, A New Kind of Science
• Wolfram Research, Wolfram Atlas of Simple Programs
• 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 (2, 1, -2).

Crossrefs

Cf. A000034, A000069, A000079, A002450, A010674, A014551, A083420, A266161.

Programs

Haskell

import Data.List (transpose)
a140253 n = a140253_list !! n
a140253_list = -1 : concat
                    (transpose [a083420_list, map (* 2) a083420_list])
-- Reinhard Zumkeller, Dec 22 2015

Maple

A140253:=proc(n): if type(n, odd) then 2*4^(((n+1)/2)-1)-1 else 2*(2*4^((n/2)-1)-1) fi: end: seq(A140253(n),n=0..29); # Johannes W. Meijer, Jun 24 2011

Mathematica

Table[(2^(n+1) - 3 + (-1)^(n+1))/2, {n, 0, 30}] (* Jean-François Alcover, Jun 05 2017 *)

Formula

a(2*n) = 2*A083420(n-1) and a(2*n+1) = A083420(n)

a(n+1) - a(n) = A014551(n); Jacobsthal-Lucas numbers.

a(2*n) + a(2*n+1) = 9*A002450(n)

a(n+1) - 2*a(n) = A010674(n+1); repeat 3, 0.

a(n) + A000034(n+1) = A000079(n); powers of 2.

a(n)= a(n-1) + 2*a(n-2) + 3. - Gary Detlefs, Jun 22 2010

a(n+1) = A000069(2^n); odious numbers. - Johannes W. Meijer, Jun 24 2011

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n>2, a(0) = -1, a(1) = 1, a(2) = 2. - Philippe Deléham, Feb 25 2012

G.f.: (x^2+3*x-1)/((1-2*x)*(1-x)*(1+x)). - Philippe Deléham, Feb 25 2012

Extensions

Edited, corrected and information added by Johannes W. Meijer, Jun 24 2011