A092896 - OEIS

%I #48 Feb 21 2021 14:08:55

%S 1,1,5,17,65,257,1025,4097,16385,65537,262145,1048577,4194305,

%T 16777217,67108865,268435457,1073741825,4294967297,17179869185,

%U 68719476737,274877906945,1099511627777,4398046511105,17592186044417,70368744177665,281474976710657

%N Related to random walks on the 4-cube.

%C 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.

%C 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

%H Vincenzo Librandi, <a href="/A092896/b092896.txt">Table of n, a(n) for n = 0..200</a>

%H M. Kac, <a href="http://www.jstor.org/stable/2304386">Random walk and the theory of Brownian motion</a>, Amer. Math. Monthly, 54:369-391, 1947.

%H Richard M. Low and Ardak Kapbasov, <a href="https://www.emis.de/journals/JIS/VOL20/Low/low2.html">Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 13.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-4).

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

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

%F a(n+1) = 4^n + 1 - 0^n = A002450(n+1) - 4*A002450(n-1). - _Paul Barry_, Mar 13 2008

%F a(n) = A052539(n-1), n > 1. - _R. J. Mathar_, Sep 08 2008

%F 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

%F a(n) = 4^(n-1) + 1 for n > 1. - _Colin Barker_, Nov 25 2016

%F E.g.f.: (exp(4*x) + 4*exp(x) - 1 - 4*x)/4. - _G. C. Greubel_, Feb 21 2021

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

%t CoefficientList[Series[(1 -4x +4x^2 -4x^3)/((1-x)(1-4x)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Feb 20 2014 *)

%t LinearRecurrence[{5,-4}, {1,1,5,17}, 30] (* _Harvey P. Dale_, Mar 19 2016 *)

%o (PARI) Vec((1-4*x+4*x^2-4*x^3)/((1-x)*(1-4*x)) + O(x^30)) \\ _Colin Barker_, Nov 25 2016

%o (Sage) [1 if n<2 else 4^(n-1) +1 for n in [0..30]]; # _G. C. Greubel_, Feb 21 2021

%o (Magma) [n lt 2 select 1 else 4^(n-1) +1: n in [0..30]]; // _G. C. Greubel_, Feb 21 2021

%Y Cf. A002450, A052539, A066443, A087289.

%Y Cf. A074515, A091775.

%K easy,nonn

%O 0,3

%A _Paul Barry_, Mar 12 2004