A008776 Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).

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

Offset: 0

Author

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

Keywords

• easy
• nonn

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

2*Sum_{n>=2} 1/A083667(n) = 2*Sum_{n>=2} 2^(-n)*3^(-((n*(n-1))/2)) = Sum_{n>=1} 1/Product_{k=1..n} A008776(k) = Sum_{n>=1} 1/Product_{k=1..n} 2*3^k = 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

a(n) is the multiplicative order of k modulo 3^(n+1), where k is any number congruent to 2 or 5 modulo 9. Note that for n > 0, k is a primitive root modulo 3^(n+1) if and only if k == 2, 5 (mod 9). - Jianing Song, Apr 20 2021

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
• Shaoshi Chen, Hanqian Fang, Sergey Kitaev, and Candice X.T. Zhang, Patterns in Multi-dimensional Permutations, arXiv:2411.02897 [math.CO], 2024. See pp. 2, 26.
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 170
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Tanya Khovanova, Recursive Sequences
• Craig Knecht, Sphinx tiling of a repetitive shape.
• 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).

Crossrefs

Apart from initial term, same as A025192.

Cf. A080643.

Cf. A000244.

Programs

GAP

List([0..30], n-> 2*3^n); # G. C. Greubel, Sep 11 2019

Haskell

a008776 = (* 2) . (3 ^)
a008776_list = iterate (* 3) 2  -- Reinhard Zumkeller, Oct 19 2015

Magma

[2*3^n: n in [0..30]]; // G. C. Greubel, Sep 11 2019

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 *)
NestList[3#&,2,50] (* Harvey P. Dale, Nov 28 2022 *)

PARI

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

Python

def A008776(n): return 3**n<<1 # Chai Wah Wu, Apr 02 2025

Sage

[2*3^n for n in (0..30)] # G. C. Greubel, Sep 11 2019

Formula

a(n) = 2*3^n.

a(n) = 3*a(n-1).

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

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

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

From Paul Curtz, Jan 20 2009: (Start)

a(n) = A048473(n) + 1.

a(n) = A052919(n+1)-1.

a(n) = A115099(n) - 2.

a(n) = A100774(n) + 2. (End)

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

Extensions

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