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A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6). 156
2, 6, 18, 54, 162, 486, 1458, 4374, 13122, 39366, 118098, 354294, 1062882, 3188646, 9565938, 28697814, 86093442, 258280326, 774840978, 2324522934, 6973568802, 20920706406, 62762119218, 188286357654, 564859072962, 1694577218886, 5083731656658, 15251194969974 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Definitions of Pisot and related sequences:

Pisot sequence E(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2) + 1/2) = nearest integer to a(n-1)^2/a(n-2), with 0<x<y.

Pisot sequence L(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2)).

Pisot sequence P(x, y): a(0) = x, a(1) = y, a(n) = ceiling(a(n-1)^2/a(n-2) - 1/2).

Pisot sequence T(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)).

Pisot/Shallit sequence S(x, y): a(0) = x, a(1) = y, a(n) = floor(a(n-1)^2/a(n-2)+1).

A025192 is the main entry for the sequence of numbers 2*3^n.

Number of tilings of a 4 X 4n+4 rectangle into T tetrominoes.

Numbers n such that 3^n = n/2 mod n. Cf. A066601 3^n mod n. - Zak Seidov, Aug 26 2006, Nov 20 2008

For n>=1, a(n) is equal to the number of functions f:{1,2...,n}->{1,2,3} such that for a fixed x in {1,2,...,n} and a fixed y in {1,2,3} we have f(x) != y. - Aleksandar M. Janjic and Milan Janjic, Mar 27 2007

a(n) = A048473+1 = A048473+A000012. a(n) = A052919(n+1)-1. a(n) = A115099-2. a(n) = A100774+2. See A007395. - Paul Curtz, Jan 20 2009

a(n+1) is the number of compositions of n when there are 2 types of each natural number. - Milan Janjic, Aug 13 2010

2*sum(1/A083667(n),n >= 2) = 2*sum(2^(-n)*3^(-((n*(n-1))/2)),n >= 2) = sum(1/prod(A008776(k),k=1..n),n >= 1 = Sum(1/(Product(2*3^k,k=1...n)),n >= 1) = 0.17609845431233461692099660022134... . - Alexander R. Povolotsky, Aug 08 2011

Number of monic squarefree polynomials over F_3 of degree n+1. - Charles R Greathouse IV, Feb 07 2012

a(n) is the sum of the elements of the n-th power of the matrix {{1, 2}, {2, 1}}. - Griffin N. Macris, Mar 25 2016

Let D(m) denote the set of divisors of a number m, and consider s1(m) and s2(m) the sums of those divisors that are congruent to 1 and 2 (mod 3) respectively. This sequence lists the numbers m such that s1(m) = 1 and s2(m) = 2. - Michel Lagneau, Feb 09 2017

REFERENCES

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 203).

LINKS

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..200

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 170

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Tanya Khovanova, Recursive Sequences

C. Moore, Some Polyomino Tilings of the Plane, arXiv:math/9905012 [math.CO], 1999.

C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scu. Norm. Sup. Pisa 2 ser, vol 7. no 3-4 (1938) p 205-248.

Index entries for linear recurrences with constant coefficients, signature (3).

Craig Knecht, Sphinx tiling of a repetitive shape.

FORMULA

a(n) = 2*3^n; a(n) = 3a(n-1).

G.f.: 2/(1-3x) . - Philippe Deléham, Oct 08 2007

a(n-1) = phi(3^n). - Artur Jasinski, Nov 19 2008

E.g.f.: 2*e^(3*x). - Mohammad K. Azarian, Jan 15 2009

If p[i]=2, (i>=1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n-1)=det A. - Milan Janjic, Apr 29 2010

G.f.: (1/2/G(0)-1)/x^2 where G(k) = 1 - 2^k/(2 - 4*x/(2*x - 2^k/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Dec 22 2012

G.f.: - G(0)/x where G(k) = 1 - 1/(1-2*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 25 2013

G.f.: (1 - 1/Q(0))/x where Q(k) = 1 - x*(2*k-2)/(1 - x*(2*k+5)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 19 2013

G.f.: W(0), where W(k) = 1 + 1/(1 - x*(2*k+3)/(x*(2*k+4) + 1/W(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Aug 28 2013

MAPLE

# E(x, y) is f(n, x, y, 1/2), T(x, y) is f(n, x, y, 0), and S(x, y) is f(n, x, y, 1).

f:=proc(n, x, y, r) option remember;

if n=0 then x

elif n=1 then y

else floor(f(n-1, x, y, r)^2/f(n-2, x, y, r) + r); fi; end;

[seq(f(n, 2, 6, 1/2), n=0..30)];

# N. J. A. Sloane, Jul 30 2016

MATHEMATICA

Table[EulerPhi[3^n], {n, 0, 100}] (* Artur Jasinski, Nov 19 2008 *)

Table[MatrixPower[{{1, 2}, {1, 2}}, n][[1]][[2]], {n, 0, 44}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2010 *)

PROG

(PARI) a(n)=3^n<<1 \\ corrected by Michel Marcus, Aug 03 2015

(Haskell)

a008776 = (* 2) . (3 ^)

a008776_list = iterate (* 3) 2  -- Reinhard Zumkeller, Oct 19 2015

CROSSREFS

Apart from initial term, same as A025192.

Cf. A080643.

Cf. A000244.

Sequence in context: A179355 A179362 A025192 * A134635 A192338 A114464

Adjacent sequences:  A008773 A008774 A008775 * A008777 A008778 A008779

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane, David W. Wilson

EXTENSIONS

Jasinski formula corrected by Charles R Greathouse IV, Feb 18 2011

STATUS

approved

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Last modified June 23 21:29 EDT 2017. Contains 288675 sequences.