

A215885


a(n) = 3*a(n1)  a(n3), with a(0) = 3, a(1) = 3, and a(2) = 9.


8



3, 3, 9, 24, 69, 198, 570, 1641, 4725, 13605, 39174, 112797, 324786, 935184, 2692755, 7753479, 22325253, 64283004, 185095533, 532961346, 1534601034, 4418707569, 12723161361, 36634883049, 105485941578, 303734663373, 874569107070, 2518221379632, 7250929475523
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OFFSET

0,1


COMMENTS

The Berndttype sequence number 5a for the argument 2Pi/9 defined by the first relation from the section "Formula". We see that a(n) is equal to the sum of the nth negative powers of the c(j) := 2*cos(2*Pi*j/9), j=1,2,4 (the A215664(n) is equal to the respective nth positive powers, further both sequences can be obtained from the twosided recurrence relation: X(n+3) = 3*X(n+1)  X(n), n in Z, with X(1) = X(0) = 3, and X(1) = 0).
From the last formula in Witula's comments to A215664 it follows that 2*(1)^n*a(n) = A215664(n)^2  A215664(2*n).
We note that abs(A215885(n)/3) = A147704(n).
The following decomposition holds true: (X  c(1)^(n))*(X  c(2)^(n))*(X  c(4)^(n)) = X^3  a(n)*X^2  (1)^n*A215664(n)*X  (1)^n.
For n >= 1, a(n) is the number of cyclic (0,1,2)compositions of n that avoid the pattern 110 provided the positions of the parts of the composition on the circle are fixed. (Similar comments hold for the pattern 012 and for the pattern 001.)  Petros Hadjicostas, Sep 13 2017
See the Maple program by Edlin and Zeilberger for counting the qary cyclic compositions of n that avoid one or more patterns provided the positions of the parts of the composition are fixed on the circle. The program is located at D. Zeilberger's personal website (see links). For the sequence here, q=3 and the pattern is A=110.  Petros Hadjicostas, Sep 13 2017


LINKS

Michael De Vlieger, Table of n, a(n) for n = 0..1999
A. E. Edlin and D. Zeilberger, The GouldenJackson cluster method for cyclic words, Adv. Appl. Math. 25 (2000), 228232.
A. E. Edlin and D. Zeilberger, Maple program.
Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
Roman Witula, Ramanujan Type Trigonometric Formulae, Demonstratio Math. 45 (2012), 779796.
Index entries for linear recurrences with constant coefficients, signature (3,0,1).


FORMULA

a(n) = c(1)^(n) + c(2)^(n) + c(4)^(n) = (c(1)*c(2))^n + (c(1)*c(4))^n + (c(2)*c(4))^n, where c(j) := 2*cos(2*Pi*j/9).
G.f.: Sum_{n>=0} a(n)*x^n = 33*x*(x^21)/(13*x+x^3) = 3*(12*x)/(13*x+x^3).
G.f. of Edlin and Zeilberger (2000): 1+Sum_{n>=1} a(n)*x^n = 13*x*(x^21)/(13*x+x^3) = (12*x^3)/(13*x+x^3).  Petros Hadjicostas, Sep 13 2017


EXAMPLE

For n=3, we have a(3) = 3^3  3 = 24 ternary cyclic compositions of n=3 (with fixed positions on the circle for the parts) that avoid 110 because we have to exclude 110, 101, and 011.  Petros Hadjicostas, Sep 13 2017


MATHEMATICA

LinearRecurrence[{3, 0, 1}, {3, 3, 9}, 50].


PROG

(PARI) x='x+O('x^99); Vec(3*(12*x)/(13*x+x^3)) \\ Altug Alkan, Sep 13 2017


CROSSREFS

Cf. A215664, A215665, A215666, A274018.
Sequence in context: A059728 A257180 A184694 * A176158 A083008 A268092
Adjacent sequences: A215882 A215883 A215884 * A215886 A215887 A215888


KEYWORD

sign,easy


AUTHOR

Roman Witula, Aug 25 2012


STATUS

approved



