

A138395


a(n) = 6*a(n1)  3*a(n2), a(1) = 1, a(2) = 6.


8



1, 6, 33, 180, 981, 5346, 29133, 158760, 865161, 4714686, 25692633, 140011740, 762992541, 4157920026, 22658542533, 123477495120, 672889343121, 3666903573366, 19982753410833, 108895809744900, 593426598236901, 3233872160186706, 17622953166409533
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OFFSET

1,2


COMMENTS

A084120(n) = 6*a(n1)  3*a(n2) but begins (1, 3, 15, 81,...).
a(n)/a(n1) tends to 3+sqrt(6) = 5.44948974...
a(n) equals the number of words of length n1 over {0,1,2,3,4,5} avoiding 01, 02 and 03.  Milan Janjic, Dec 17 2015


LINKS

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (6, 3).


FORMULA

a(n) = ((3+sqrt(6))^n  (3sqrt(6))^n)/(2*sqrt(6)).  Alexander R. Povolotsky, Apr 01 2008
a(n) = lower left term of nth power of 2 X 2 matrix [1,2; 1,5].
G.f.: 1/(1  6x + 3x^2).  Philippe Deléham, Sep 09 2009
a(n) = Chebyshev_U(n, sqrt(3))*(sqrt(3))^n.  Paul Barry, Sep 28 2009


EXAMPLE

a(5) = 981 = 6*a(4)  3*a(3) = 6*180  3*33.


MATHEMATICA

a[n_]:=(MatrixPower[{{1, 2}, {1, 5}}, n].{{1}, {1}})[[2, 1]]; Table[a[n], {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2010 *)
LinearRecurrence[{6, 3}, {1, 6}, 30] (* Harvey P. Dale, Jan 18 2012 *)


PROG

(MAGMA) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n1)3*Self(n2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015
(PARI) Vec(1/(16*x+3*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015


CROSSREFS

Cf. A084120.
Sequence in context: A111994 A092851 A137627 * A050151 A009162 A012718
Adjacent sequences: A138392 A138393 A138394 * A138396 A138397 A138398


KEYWORD

nonn,easy


AUTHOR

Gary W. Adamson, Mar 19 2008


EXTENSIONS

More terms from Philippe Deléham, Sep 09 2009
a(21) and first formula corrected by Klaus Brockhaus, Oct 05 2009
Extended by T. D. Noe, May 23 2011


STATUS

approved



