OFFSET
0,3
COMMENTS
The inverse binomial transform is 1, 1, 4, -2, 10, -14, 34, -62 which leads to (-1)^(n+1)*A135440(n).
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 linear recurrences with constant coefficients, signature (2, 1, -2).
FORMULA
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)= 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
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 *)
PROG
(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
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Jun 23 2008
EXTENSIONS
Edited, corrected and information added by Johannes W. Meijer, Jun 24 2011
STATUS
approved