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A052533
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Expansion of (1-x)/(1-x-3x^2).
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4
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1, 0, 3, 3, 12, 21, 57, 120, 291, 651, 1524, 3477, 8049, 18480, 42627, 98067, 225948, 520149, 1197993, 2758440, 6352419, 14627739, 33684996, 77568213, 178623201, 411327840, 947197443, 2181180963, 5022773292, 11566316181, 26634636057
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OFFSET
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0,3
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COMMENTS
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Form the graph with matrix A=[0,1,1,1;1,1,0,0;1,0,1,0;1,0,0,1]. A052533 counts closed walks of length n at the vertex without loop. - Paul Barry, Oct 02 2004
Let M = [0, sqrt(3); sqrt(3), 1] be a 2X2 matrix. Then A052533={[M^n]_(1,1)}. Note also that {[M^n]_(2,2)}=A006130. -- L. Edson Jeffery, Nov 25, 2011.
Pisano period lengths: 1, 3, 1, 6, 24, 3, 24, 6, 1, 24,120, 6,156, 24, 24, 12, 16, 3, 90, 24,.. - R. J. Mathar, Aug 10 2012
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LINKS
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Table of n, a(n) for n=0..30.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 463
Index to sequences with linear recurrences with constant coefficients, signature (1,3).
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FORMULA
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G.f.: (1-x)/(1-x-3*x^2).
a(n) = A006130(n)-A006130(n-1).
Recurrence: {a(1)=0, a(0)=1, 3*a(n)+a(n+1)-a(n+2)=0}.
Sum(1/13*(-1+7*_alpha)*_alpha^(-1-n), _alpha=RootOf(-1+_Z+3*_Z^2)).
a(n) = sum{k=0..floor(n/2), C(n-k-1,n-2k)*3^k}. [From Paul Barry, Mar 16 2010]
If p[1]=0, and p[i]=3, (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)=det A. [From Milan R. Janjic, Apr 29 2010]
a(n) = (1/2)*{[(1/2)*(1-*sqrt(13))]^n+[(1/2)*(1+*sqrt(13))]^n}+(1/26)*sqrt(13*{[(1/2)*(1-*sqrt(13))]^n)-[(1/2)*(1+*sqrt(13))]^n}, with n>=0. [From Paolo P. Lava, May 10 2010]
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MAPLE
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spec := [S, {S=Sequence(Prod(Z, Union(Z, Z, Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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CROSSREFS
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Cf. A006130.
Sequence in context: A075780 A078666 A006804 * A136533 A192307 A161804
Adjacent sequences: A052530 A052531 A052532 * A052534 A052535 A052536
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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EXTENSIONS
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More terms from James A. Sellers, Jun 06 2000
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STATUS
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approved
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