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) = 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
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
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Mar 12 2004
STATUS
approved