A246036 Expansion of (1+4*x)/((1+2*x)*(1-4*x)).
1, 6, 20, 88, 336, 1376, 5440, 21888, 87296, 349696, 1397760, 5593088, 22368256, 89481216, 357908480, 1431666688, 5726601216, 22906535936, 91625881600, 366504050688, 1466015154176, 5864062713856, 23456246661120, 93824995033088, 375299963355136, 1501199886974976, 6004799480791040, 24019198057381888
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- Christophe Doche, Even moments of generalized Rudin-Shapiro polynomials, Mathematics of computation 74.252 (2005): 1923-1935.
- Christophe Doche and Laurent Habsieger, Moments of the Rudin-Shapiro polynomials, Journal of Fourier Analysis and Applications 10.5 (2004): 497-505.
- Shalosh B. Ekhad, Explicit Generating Functions, Asymptotics, and More for the First 10 Even Moments of the Rudin-Shapiro Polynomials, Preprint, 2016.
- Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata, arXiv:1503.01796 [math.CO], 2015; see also the Accompanying Maple Package.
- Shalosh B. Ekhad, N. J. A. Sloane, and Doron Zeilberger, Odd-Rule Cellular Automata on the Square Grid, arXiv:1503.04249 [math.CO], 2015.
- N. J. A. Sloane, On the No. of ON Cells in Cellular Automata, Video of talk in Doron Zeilberger's Experimental Math Seminar at Rutgers University, Feb. 05 2015: Part 1, Part 2
- N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.
- Index entries for sequences related to cellular automata
- Index entries for linear recurrences with constant coefficients, signature (2,8).
Programs
-
Magma
I:=[1,6]; [n le 2 select I[n] else 2*Self(n-1)+8*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 22 2014
-
Mathematica
CoefficientList[Series[(1+4x)/((1+2x)(1-4x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2014 *) -
PARI
Vec((1+4*x)/((1+2*x)*(1-4*x)) + O(x^100)) \\ Colin Barker, Aug 22 2014
-
PARI
apply( A246036(n)=(4^(1+n)-(-2)^n)/3, [0..30]) \\ M. F. Hasler, Sep 18 2020
-
SageMath
A246036= BinaryRecurrenceSequence(2,8,1,6) [A246036(n) for n in range(41)] # G. C. Greubel, Mar 08 2023
Formula
a(n) = 2*a(n-1) + 8*a(n-2).
a(n) = (4^(1+n) - (-2)^n)/3. - Colin Barker, Aug 22 2014
a(n) = A054881(n+3)/8. - L. Edson Jeffery, Apr 22 2015
a(n) = A003683(n+2)/2 and the above formula follow from the explicit expression for a(n), cf. second formula. - M. F. Hasler, Sep 11 2020
a(n) = 2^n*A001045(n+2). - R. J. Mathar, Mar 08 2021
Comments