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
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).
Programs
-
Magma
[n lt 2 select 1 else 4^(n-1) +1: n in [0..30]]; // 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 *) -
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
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) = 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
Comments