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A119826 Number of ternary words of length n with no 000's. 6
1, 3, 9, 26, 76, 222, 648, 1892, 5524, 16128, 47088, 137480, 401392, 1171920, 3421584, 9989792, 29166592, 85155936, 248624640, 725894336, 2119349824, 6187737600, 18065963520, 52746101888, 153999606016, 449623342848, 1312738101504, 3832722100736, 11190167090176, 32671254584832 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Column 0 of A119825.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..700

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

FORMULA

G.f. (1+z+z^2)/(1-2*z-2*z^2-2*z^3).

a(n-1) = sum(m=1..n, sum(k=m..n, C(k-1,m-1) * sum(j=0..k, C(j,n-3*k+2*j) * C(k,j)))) [From Vladimir Kruchinin, Apr 25 2011]

G.f. for sequence with 1 prepended: 1/( 1 - sum(k>=1,  (x+x^2+x^3)^k ) ). [Joerg Arndt, Sep 30 2012]

a(n) = round(3/2*((r+s+2)/3)^(n+3)/(r^2+s^2+10)), where r=(53+3*sqrt(201))^(1/3), s=(53-3*sqrt(201))^(1/3); r and s are the real roots of the polynomial x^6 - 106*x^3 + 1000. [Anton Nikonov, Jul 11 2013]

a(n) = A077835(n)+A077835(n-1)+A077835(n-2). - R. J. Mathar, Aug 07 2015

EXAMPLE

a(4)=76 because among the 3^4=81 ternary words of length 4 only 0000, 0001, 0002, 1000 and 2000 contain 000's.

MAPLE

g:=(1+z+z^2)/(1-2*z-2*z^2-2*z^3): gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=0..28);

# second Maple program

a:= n-> (<<0|1|0>, <0|0|1>, <2|2|2>>^n. <<1, 3, 9>>)[1, 1]:

seq (a(n), n=0..30);  # Alois P. Heinz, Oct 30 2012

MATHEMATICA

nn=30; CoefficientList[Series[(1-x^3)/(1-3x+2x^4), {x, 0, nn}], x]  (* Geoffrey Critzer, Oct 30 2012 *)

LinearRecurrence[{2, 2, 2}, {1, 3, 9}, 30] (* Jean-Fran├žois Alcover, Dec 25 2015 *)

PROG

(Maxima)

a(n):=sum(sum(binomial(k-1, m-1)*sum(binomial(j, n-3*k+2*j)*binomial(k, j), j, 0, k), k, m, n), m, 1, n); \\ Vladimir Kruchinin, Apr 25 2011

CROSSREFS

Cf. A119825, A119827.

Sequence in context: A005774 A273343 A101169 * A027915 A295115 A114982

Adjacent sequences:  A119823 A119824 A119825 * A119827 A119828 A119829

KEYWORD

nonn,easy

AUTHOR

Emeric Deutsch, May 26 2006

STATUS

approved

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Last modified December 17 07:54 EST 2018. Contains 318192 sequences. (Running on oeis4.)