A092896 Related to random walks on the 4-cube.

1, 1, 5, 17, 65, 257, 1025, 4097, 16385, 65537, 262145, 1048577, 4194305, 16777217, 67108865, 268435457, 1073741825, 4294967297, 17179869185, 68719476737, 274877906945, 1099511627777, 4398046511105, 17592186044417, 70368744177665, 281474976710657

Offset

0,3

Comments

Gives the denominators in the probability that a random walk on the 4-cube returns to its starting corner on the 2n-th step. Partial sums of A092898. Binomial transform of A092897.

Palindromic numbers in base 2 with an odd number of bits that can be written as 2^(2n) + 1, n >= 1. Palindromic numbers in base 2 with an even number of bits that can be written as 2^(2n+1) + 1 are A087289. - Brad Clardy, Feb 18 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

M. Kac, Random walk and the theory of Brownian motion, Amer. Math. Monthly, 54:369-391, 1947.

Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 13.

Index entries for linear recurrences with constant coefficients, signature (5,-4).

Formula

G.f.: (1 - 4*x + 4*x^2 - 4*x^3)/((1-x)*(1-4*x)).

a(n) = 1 + 4^n/4 - 0^n/4 + Sum_{k=0..n} binomial(n, k)*k*(-1)^k.

a(n+1) = 4^n + 1 - 0^n = A002450(n+1) - 4*A002450(n-1). - Paul Barry, Mar 13 2008

a(n) = A052539(n-1), n > 1. - R. J. Mathar, Sep 08 2008

Dropping a(0) and interleaving the terms with zeros gives a sequence with e.g.f. (sin(5ix/2)/sin(ix/2) - 3)/2 = cos(2ix) + cos(ix) - 1. Similar expressions apply to A091775 and A074515, which are also power sums representable by the Bernoulli polynomials. - Tom Copeland, Oct 22 2008

a(n) = 4^(n-1) + 1 for n > 1. - Colin Barker, Nov 25 2016

E.g.f.: (exp(4*x) + 4*exp(x) - 1 - 4*x)/4. - G. C. Greubel, Feb 21 2021

Maple

A092896:= n -> `if`(n<2, 1, 4^(n-1) +1); seq(A092896(n), n = 0..30); # G. C. Greubel, Feb 21 2021

Mathematica

CoefficientList[Series[(1 -4x +4x^2 -4x^3)/((1-x)(1-4x)), {x, 0, 40}], x] (* Vincenzo Librandi, Feb 20 2014 *)
LinearRecurrence[{5, -4}, {1, 1, 5, 17}, 30] (* Harvey P. Dale, Mar 19 2016 *)

Prog

(PARI) Vec((1-4*x+4*x^2-4*x^3)/((1-x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Nov 25 2016
(Sage) [1 if n<2 else 4^(n-1) +1 for n in [0..30]]; # G. C. Greubel, Feb 21 2021
(Magma) [n lt 2 select 1 else 4^(n-1) +1: n in [0..30]]; // G. C. Greubel, Feb 21 2021

Crossrefs

Cf. A002450, A052539, A066443, A087289.

Cf. A074515, A091775.

Sequence in context: A149672 A149673 A046231 * A149674 A149675 A149676

Adjacent sequences: A092893 A092894 A092895 * A092897 A092898 A092899

Keyword

easy,nonn

Author

Paul Barry, Mar 12 2004

Status

approved