login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A282087 Number of length-n ternary strings that do not contain both "00" and "11". 1
1, 3, 9, 27, 79, 229, 657, 1871, 5295, 14909, 41801, 116783, 325287, 903741, 2505377, 6932479, 19151519, 52833853, 145578265, 400705135, 1101936119, 3027902045, 8314284721, 22816209855, 62579270191, 171559358493, 470132335209, 1287861941487, 3526800739399 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Math StackExchangeFind a recurrence relation for the number of ternary strings of length ? that do not contain two consecutive 0s and two consecutive 1s

Index entries for linear recurrences with constant coefficients, signature (4,-1,-6,-2).

FORMULA

a(n) = 4*a(n-1) - a(n-2) - 6*a(n-3) - 2*a(n-4) for n >= 4 (derived in the math.stackexchange.com link).

From Colin Barker, Feb 07 2017: (Start)

a(n) = -(1-u)^(1+n)/2 - (1+u)^(1+n)/2 + (1-v)^n - (2*(1-v)^n)/v + (1+v)^n + (2*(1+v)^n) / v where u=sqrt(2) and v=sqrt(3).

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

(End)

EXAMPLE

for n=5 the 229 acceptable ternary strings are all length 5 strings of '0', '1', and '2' _except_ '00011', '00110', '00111', '00112', '00211', '01100', '10011', '11000', '11001', '11002', '11100', '11200', '20011', '21100'.

MATHEMATICA

Table[3^n, {n, 0, 3}]~Join~LinearRecurrence[{4, -1, -6, -2}, {79, 229, 657, 1871}, 24] (* or *)

Table[Count[Tuples[Range[0, 2], n], w_ /; Boole[SequenceCount[w, {0, 0}] > 0] Boole[SequenceCount[w, {1, 1}] > 0] == 0], {n, 0, 12}] (* Michael De Vlieger, Feb 05 2017, latter program version 10.1 *)

PROG

(Python)

import itertools

# Not feasible on most machines for large numbers

def find_a_sub_n(n):

    c = 0

    for q in itertools.product(*([['0', '1', '2']]*n)):

        h = ''.join(q)

        if not (('11' in h) and ('00' in h)):

            c = c+1

    return c

(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -2, -6, -1, 4]^n*[1; 3; 9; 27])[1, 1] \\ Charles R Greathouse IV, Feb 05 2017

(PARI) Vec((1 + x)*(1 - 2*x) / ((1 - 2*x - x^2)*(1 - 2*x - 2*x^2)) + O(x^30)) \\ Colin Barker, Feb 07 2017

CROSSREFS

Cf. A078057 (number of length-n ternary strings that contain neither "00" nor "11").

Sequence in context: A269650 A266497 A291020 * A238440 A269578 A026289

Adjacent sequences:  A282084 A282085 A282086 * A282088 A282089 A282090

KEYWORD

nonn,easy

AUTHOR

Daniel T. Martin, Feb 05 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 19 18:51 EDT 2019. Contains 326133 sequences. (Running on oeis4.)