A087206 a(n) = 2*a(n-1) + 4*a(n-2); with a(0)=1, a(1)=4.

1, 4, 12, 40, 128, 416, 1344, 4352, 14080, 45568, 147456, 477184, 1544192, 4997120, 16171008, 52330496, 169345024, 548012032, 1773404160, 5738856448, 18571329536, 60098084864, 194481487872, 629355315200, 2036636581888, 6590694424576, 21327935176704, 69018648051712

Offset

0,2

Comments

Binomial transform of A056487.

Unsigned version of A152174.

Number of words of length n over the alphabet {1,2,3,4} such that no odd letter is followed by an odd letter. - Armend Shabani, Feb 18 2017

From Sean A. Irvine, Jun 06 2025: (Start)

Also, the number of walks of length n starting at 0 in the following graph:

1---2

|\ /|

| 0 |

|/ \|

4---3. (End)

Links

Table of n, a(n) for n=0..27.

Jens Christian Claussen, Time-evolution of the Rule 150 cellular automaton activity from a Fibonacci iteration, arXiv:math/0410429 [math.CO], 2004. See Table II, p. 4.

Sean A. Irvine, Walks on Graphs.

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015.

Index entries for linear recurrences with constant coefficients, signature (2,4).

Formula

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

a(n) = (1/2) * A063727(n+1).

a(n) = (1-sqrt(5))^n*(1/2-3*sqrt(5)/10)+(1+sqrt(5))^n*(1/2+3*sqrt(5)/10).

a(n) = 2^n*Fibonacci(n+2). - Paul Barry, Mar 22 2004

a(n) = ((1+sqrt(5))^n-(1-sqrt(5))^n)/sqrt(80). Offset 2. a(4)=12. - Al Hakanson (hawkuu(AT)gmail.com), Apr 11 2009

G.f.: 1/(-2*x-1/(-2*x-1)). - Paul Barry, Mar 24 2010

E.g.f.: exp(x) * (cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5)). - Amiram Eldar, Feb 12 2026

Mathematica

LinearRecurrence[{2, 4}, {1, 4}, 25] (* Jean-François Alcover, Sep 21 2017 *)

Crossrefs

Cf. A000045, A006483, A056487, A060925, A063727, A063782, A152174, A253064.

Sequence in context: A341990 A090576 A152174 * A275863 A289653 A081875

Adjacent sequences: A087203 A087204 A087205 * A087207 A087208 A087209

Keyword

easy,nonn

Author

Paul Barry, Aug 25 2003

Extensions

Comment corrected by Philippe Deléham, Nov 27 2008

Status

approved