A080674 a(n) = (4/3)*(4^n - 1).

0, 4, 20, 84, 340, 1364, 5460, 21844, 87380, 349524, 1398100, 5592404, 22369620, 89478484, 357913940, 1431655764, 5726623060, 22906492244, 91625968980, 366503875924, 1466015503700, 5864062014804, 23456248059220, 93824992236884, 375299968947540, 1501199875790164

Offset: 0

Author

N. J. A. Sloane, Mar 02 2003

Keywords

• nonn
• easy

Comments

a(n) is the number of steps which are made when generating all n-step random walks that begin in a given point P on a two-dimensional square lattice. To make one step means to move along one edge on the lattice. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Mar 10 2005

Conjectured to be the number of integers from 0 to (10^n)-1 that lack 0, 1, 2, 3, 4 and 5 as a digit. - Alexandre Wajnberg, Apr 25 2005

Gives the values of m such that binomial(4*m + 4,m) is odd. Cf. A002450, A020988 and A263132. - Peter Bala, Oct 11 2015

Also the partial sums of 4^n for n>0, cf. A000302. - Robert G. Wilson v, Sep 18 2016

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..170
• Peter Bala, A characterization of A002450, A020988 and A080674.
• Mattia Fregola, Cellular Automata RULE13 generating OEIS sequence A080674
• Index entries for linear recurrences with constant coefficients, signature (5,-4).

Crossrefs

Row n = 4 of A228275.

Cf. A000301, A000302, A002450, A020988, A024036, A080674, A263132.

Programs

Magma

[(4/3)*(4^n-1): n in [0..40] ]; // Vincenzo Librandi, Apr 28 2011

Mathematica

Table[4*(4^n-1)/3,{n,0,100}]  (* Vladimir Joseph Stephan Orlovsky, Jan 30 2012 *)
LinearRecurrence[{5,-4},{0,4},40] (* Harvey P. Dale, May 05 2018 *)

PARI

vector(100, n, n--; (4/3)*(4^n-1)) \\ Altug Alkan, Oct 11 2015

PARI

Vec(4*x/((x-1)*(4*x-1)) + O(x^40)) \\ Colin Barker, Oct 12 2015

Formula

a(n) = 2*A020988(n) = A002450(n+1) - 1 = 4*A002450(n).

a(n) = Sum_{i = 1..n} 4^i. - Adam McDougall (mcdougal(AT)stolaf.edu), Sep 29 2004

a(n) = 4*a(n-1) + 4. - Alexandre Wajnberg, Apr 25 2005

a(n) = 4^n + a(n-1) (with a(0) = 0). - Vincenzo Librandi, Aug 08 2010

From Colin Barker, Oct 12 2015: (Start)

a(n) = 5*a(n-1) - 4*a(n-2).

G.f.: 4*x / ((x-1)*(4*x-1)). (End)

E.g.f.: 4*exp(x)*(exp(3*x) - 1)/3. - Elmo R. Oliveira, Dec 17 2023