A166956 a(n) = 2^n +(-1)^n - 2.

0, -1, 3, 5, 15, 29, 63, 125, 255, 509, 1023, 2045, 4095, 8189, 16383, 32765, 65535, 131069, 262143, 524285, 1048575, 2097149, 4194303, 8388605, 16777215, 33554429, 67108863, 134217725, 268435455, 536870909, 1073741823, 2147483645, 4294967295, 8589934589

Offset: 0

Author

Paul Curtz, Oct 25 2009

Keywords

• sign
• easy

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 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. A166920, A166956, A290660, A290661, A290662.

Programs

Magma

[2^n-2+(-1)^n: n in [0..40]]; // Vincenzo Librandi, Apr 28 2011

Mathematica

LinearRecurrence[{2,1,-2},{0,-1,3},20] (* G. C. Greubel, May 29 2016 *)

Formula

a(n) = A000079(n) - A010684(n).

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

Extensions

Edited and extended by R. J. Mathar, Mar 02 2010