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Search: id:A052547
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| A052547 |
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A simple regular expression. |
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+0
5
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| 1, 0, 2, 1, 5, 5, 14, 19, 42, 66, 131, 221, 417, 728, 1341, 2380, 4334, 7753, 14041, 25213, 45542, 81927, 147798, 266110, 479779, 864201, 1557649, 2806272, 5057369, 9112264, 16420730, 29587889, 53317085, 96072133, 173118414, 311945595
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Form the graph with matrix A=[0,1,1;1,0,0;1,0,1] (P_3 with a loop at an extremity). Then A052547 counts closed walks of length n at the degree 2 vertex. - Paul Barry (pbarry(AT)wit.ie), Oct 02 2004
The characteristic polynomial x^3 - x^2 - 2*x + 1 generates a 3 step recursion: a(0)=1,a(1)=0,a(2)=2, for n>2 a(n)=a(n-1)+2*a(n-2)-a(n-3) so we can also prepend the term 1,0 to a(n) and get the same sequence, i.e. start with a(0)=1,a(1)=0,a(2)=1. - Lambert Klasen (lambert.klasen(AT)gmx.net), Jan 30 2005
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LINKS
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INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 483
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FORMULA
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a(n)=5a(n-2)-6a(n-4)+a(n-6). - Floor van Lamoen (fvlamoen(AT)hotmail.com), Nov 02 2005
G.f.: -(-1+x)/(x^3-2*x^2-x+1)
Recurrence: {a(1)=0, a(0)=1, a(2)=2, a(n)-2*a(n+1)-a(n+2)+a(n+3)}
Sum(-1/7*_alpha*(-3+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-2*_Z^2-_Z+1))
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MAPLE
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spec := [S, {S=Sequence(Prod(Z, Union(Z, Prod(Z, Sequence(Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
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PROGRAM
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(PARI) a(n)=if(n==0, 1, if(n==1, 0, if(n==2, 1, a(n-1)+2*a(n-2)-a(n-3)))) for(i=0, 20, print1(a(i), ", ")) (Klasen)
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CROSSREFS
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Cf. A096976, A028495.
Sequence in context: A032006 A167158 A074392 this_sequence A096976 A119245 A128731
Adjacent sequences: A052544 A052545 A052546 this_sequence A052548 A052549 A052550
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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 (sellersj(AT)math.psu.edu), Jun 05 2000
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