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A008776
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Pisot sequences E(2,6), L(2,6), P(2,6), T(2,6).
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118
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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; internal format)
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OFFSET
| 0,1
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COMMENTS
| A025192 is the main entry for this sequence.
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 R. Janjic (agnus(AT)blic.net), 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. [From Paul Curtz (bpcrtz(AT)free.fr), Jan 20 2009]
a(n+1) is the number of compositions of n when there are 2 types of each natural number. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
2*sum(1/A083667(n),n=2...infinity) = 2*sum(2^(-n)*3^(-((n*(n-1))/2)),n=2...infinity) = sum(1/prod(A008776(k),k=1..n),n=1...infinity = Sum(1/(Product(2*3^k,k=1...n)),n=1...infinity) = 0.17609845431233461692099660022134.... [From 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]
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REFERENCES
| S. J. Cyvin and I. Gutman, Kekule structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 203).
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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, [math/9905012] Some Polyomino Tilings of the Plane
Index entries for sequences related to linear recurrences with constant coefficients, signature (3).
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FORMULA
| a(n) = 2*3^n; a(n) = 3a(n-1).
Pisot sequence E(x, y): a(0) = x, a(1) = y, a(n) = roundUp(a(n-1)^2/a(n-2)) = [ a(n-1)^2/a(n-2) + 1/2 ].
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) = roundDown(a(n-1)^2/a(n-2)) = 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)) = [ a(n-1)^2/a(n-2) ].
G.f.: 2/(1-3x) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 08 2007
a(n-1)=phi(3^n) [From Artur Jasinski (grafix(AT)csl.pl), Nov 19 2008]
E.g.f.: 2*e^(3*x). [From Mohammad K. Azarian (azarian(AT)evansville.edu), 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. [From Milan R. Janjic (agnus(AT)blic.net), Apr 29 2010]
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MATHEMATICA
| Table[EulerPhi[3^n], {n, 0, 100}] [From Artur Jasinski (grafix(AT)csl.pl), Nov 19 2008]
Table[MatrixPower[{{1, 2}, {1, 2}}, n][[1]][[2]], {n, 0, 44}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 20 2010]
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PROG
| (PARI) a(n)=3^(n-1)<<1
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CROSSREFS
| Apart from initial term, same as A025192. Cf. A080643.
Sequence in context: A179355 A179362 A025192 * A134635 A192338 A114464
Adjacent sequences: A008773 A008774 A008775 * A008777 A008778 A008779
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KEYWORD
| easy,nonn,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| Jasinski formula corrected by Charles R Greathouse IV, Feb 18 2011
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