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

0,2

Comments

Also, fourth moments of Rudin-Shapiro polynomials (see Doche, Doche-Habsieger, Ekhad papers). - Doron Zeilberger, Apr 15 2016

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).

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

Mathematica

CoefficientList[Series[(1+4x)/((1+2x)(1-4x)), {x, 0, 50}], x] (* Vincenzo Librandi, Aug 22 2014 *)

Prog

(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
(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

Crossrefs

Cf. A001045, A003683, A054881, A246037, A271494, A271495, A271496.

Sequence in context: A255469 A226638 A274071 * A151485 A191424 A333048

Adjacent sequences: A246033 A246034 A246035 * A246037 A246038 A246039

Keyword

nonn,easy

Author

N. J. A. Sloane, Aug 21 2014

Status

approved