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A334656 a(n) is the number of words of length n on the alphabet {0,1,2} with the number of 0's plus the number of 1's congruent to the number of 2's modulo 3. 1
1, 0, 4, 9, 24, 90, 225, 756, 2160, 6561, 19764, 58806, 177633, 530712, 1595052, 4782969, 14346720, 43053282, 129127041, 387440172, 1162241784, 3486784401, 10460412252, 31380882462, 94143533121, 282429005040, 847289140884, 2541865828329, 7625595890664 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

Thomas A. Sudkamp, An Introduction to Languages and Machines. second edition 1997 Addison-Wesley.

LINKS

Table of n, a(n) for n=0..28.

Brallan S. Rueda Mantilla, Automata, regular expression and recurrence law for a(n)

Index entries for linear recurrences with constant coefficients, signature (0,6,9).

FORMULA

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

a(n) = 6*a(n-2) + 9*a(n-3).

G.f.: (1 - 2*x^2)/((1 - 3*x)*(1 + 3*x + 3*x^2)). - Andrew Howroyd, Sep 11 2020

E.g.f.: (exp(3*x) + 2*exp(-3*x/2)*cos(sqrt(3)*x/2))/3. - Stefano Spezia, Sep 11 2020

EXAMPLE

The a(3)=9 words are (0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1), (2,2,2).

MATHEMATICA

CoefficientList[Series[(1 - 2 x^2)/((1 - 3 x) (1 + 3 x + 3 x^2)), {x, 0, 28}], x] (* Michael De Vlieger, Sep 11 2020 *)

PROG

(PARI) Vec((1 - 2*x^2)/((1 - 3*x)*(1 + 3*x + 3*x^2)) + O(x^30)) \\ Andrew Howroyd, Sep 11 2020

CROSSREFS

Cf. A000244 (3^n).

Sequence in context: A081149 A059507 A128416 * A175121 A133905 A131826

Adjacent sequences:  A334653 A334654 A334655 * A334657 A334658 A334659

KEYWORD

nonn,easy

AUTHOR

Brallan S. Rueda Mantilla, Sep 10 2020

STATUS

approved

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Last modified November 30 09:54 EST 2021. Contains 349419 sequences. (Running on oeis4.)