A074515 -id:A074515 - cp's OEIS frontend

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

Paul Barry, Mar 12 2004

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

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).

Cf. A002450, A052539, A066443, A087289.

Cf. A074515, A091775.

[n lt 2 select 1 else 4^(n-1) +1: n in [0..30]]; // G. C. Greubel, Feb 21 2021

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

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 *)

Vec((1-4*x+4*x^2-4*x^3)/((1-x)*(1-4*x)) + O(x^30)) \\ Colin Barker, Nov 25 2016

[1 if n<2 else 4^(n-1) +1 for n in [0..30]]; # G. C. Greubel, Feb 21 2021

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

A091775 1 + 4^n + 9^n + 16^n.

Original entry on oeis.org

30, 354, 4890, 72354, 1108650, 17312754, 273234810, 4338079554, 69107159370, 1102999460754, 17623571298330, 281757423024354, 4506141560307690, 72080471098818354, 1153127396812683450, 18448597098193370754, 295164582378232361610, 4722516577573661689554

Offset: 1

Benoit Cloitre, Mar 06 2004

Index entries for linear recurrences with constant coefficients, signature (30, -273, 820, -576).

Cf. A052539, A074515.

[1+4^n+9^n+16^n : n in [1..20]]; // Wesley Ivan Hurt, Nov 26 2014

A091775:=n->1+4^n+9^n+16^n: seq(A091775(n), n=1..20); # Wesley Ivan Hurt, Nov 26 2014

Table[1 + 4^n + 9^n + 16^n, {n, 20}] (* or *) LinearRecurrence[ {30, -273, 820, -576}, {30, 354, 4890, 72354}, 20] (* Harvey P. Dale, May 04 2011 *)

For n > 0, a(n) = 5^(2*n+1)/(2*n+1)*sum(k = 0, 2*n + 1, (1/5)^k*C(2*n + 1, k)*B(k)) where B(k) is the k-th Bernoulli number.
G.f.: x*(16/(1 - 16*x) + 9/(1 - 9*x) + 4/(1 - 4*x) + 1/(1 - x)). - Harvey P. Dale, May 04 2011
a(1) = 30, a(2) = 354, a(3) = 4890, a(4) = 72354, a(n) = 30*a(n-1) - 273*a(n-2) + 820*a(n-3) - 576*a(n-4). - Harvey P. Dale, May 04 2011

Corrected and extended by Harvey P. Dale, May 04 2011

cp's OEIS Frontend

A092896 Related to random walks on the 4-cube.

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