OFFSET
0,3
COMMENTS
The inverse binomial transform yields 0,-1,5,-7,17,-31,..., a sign alternating variant of A014551.
In a table of a(n) and higher-order differences in successive rows, the main diagonal contains 0, 4, 8, 16, ... (zero followed by A020707).
Similar to 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 899", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero, which begins with 1,3,5,15,29,63,125. - Robert Price, Aug 08 2017
REFERENCES
S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..240
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) = 2*a(n-1) + a(n-2) - 2*a(n-3).
G.f.: x*(5*x -1)/((1-x)*(1-2*x)*(1+x)).
E.g.f.: exp(2*x) - 2*exp(x) + exp(-x). - G. C. Greubel, May 29 2016
MATHEMATICA
LinearRecurrence[{2, 1, -2}, {0, -1, 3}, 20] (* G. C. Greubel, May 29 2016 *)
PROG
(Magma) [2^n-2+(-1)^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul Curtz, Oct 25 2009
EXTENSIONS
Edited and extended by R. J. Mathar, Mar 02 2010
STATUS
approved