A138395 a(n) = 6*a(n-1) - 3*a(n-2), a(1) = 1, a(2) = 6.

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

Offset: 1

Author

Gary W. Adamson, Mar 19 2008

Keywords

• nonn
• easy

Comments

a(n) equals the number of words of length n-1 over {0,1,2,3,4,5} avoiding 01, 02 and 03. - Milan Janjic, Dec 17 2015

Examples

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

Links

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

Crossrefs

Cf. A084120, A190958.

Programs

Magma

I:=[1,6]; [n le 2 select I[n] else 6*Self(n-1)-3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Dec 17 2015

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 *)

PARI

Vec(1/(1-6*x+3*x^2) + O(x^100)) \\ Altug Alkan, Dec 17 2015

SageMath

A138395=BinaryRecurrenceSequence(6,-3,0,1)
[A138395(n) for n in range(1,30)] # G. C. Greubel, Jan 10 2024

Formula

Limit_{n->oo} a(n)/a(n-1) = 3 + sqrt(6) = 5.44948974...

a(n) = ((3+sqrt(6))^n - (3-sqrt(6))^n)/(2*sqrt(6)). - Alexander R. Povolotsky, Apr 01 2008

a(n) = lower left term of n-th power of 2 X 2 matrix [1,2; 1,5].

G.f.: 1/(1 - 6*x + 3*x^2). - Philippe Deléham, Sep 09 2009

a(n) = Chebyshev_U(n, sqrt(3))*(sqrt(3))^n. - Paul Barry, Sep 28 2009

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