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A052547
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Expansion of (1-x)/(1-x-2*x^2+x^3).
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20
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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, 562110290, 1012883066
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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;1,0,0;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, 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
The length of the diagonals (including the side) of a regular 7-gon (heptagon) inscribed in a circle of radius r=1 are d_1=2*sin(Pi/7) (the side length), d_2=2*cos(Pi/7)*d_1, and d_3=2*sin(3*Pi/7). The two ratios are rho := R_2 = d_2/d_1 = 2*cos(Pi/7) approximately 1.801937736, and sigma:= R_3 = d_3/d_1 = S(2,rho) = rho^2-1, approximately 2.246979604. See A049310 for Chebyshev S-polynomials. See the Steinbach reference where the basis <1,rho,sigma> has been considered for an extension of the rational field Q, which is there called Q(rho). This rho is the largest zero of S(6,x). For nonnegative powers of rho one has rho^n = C(n)*1 + B(n)*rho + A(n)*sigma, with B(n)=a(n-1), a(-1):=0, a(-2):=1, A(n)=B(n+1)-B(n-1)= A006053(n), and C(n)=B(n-1)=a(n-2), n>=0. For negative powers see A106803 and -A006054. For nonnegative and negative powers of sigma see A006054, A106803 and a(n), -A006053, respectively.
a(n) appears also in the formula for the nonpositive powers of sigma (see the comment above for the definition and the Steinbach basis) as sigma^(-n) = a(n)*1 - A006053(n+1)*rho - a(n-1)*sigma, n>=0. Put a(-1):=0. 1/sigma=sigma-rho, the smallest positive zero of S(6,x) (see A049310 for Chebyshev S-polynomials). - Wolfdieter Lang, Dec 01 2010
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
M. Gao and J. Wu, Homology Decompositions of the Loops on 1-Stunted Borel Constructions of C_2-Actions, arXiv:1301.0864 [math.AT], 2013
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 483
P. Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), no. 1, 22-31.
Index entries for linear recurrences with constant coefficients, signature (1,2,-1).
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FORMULA
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a(n) = a(n-1) + 2*a(n-2) - a(n-3), with a(0)=1, a(1)=0, a(2)=2.
a(n) = Sum(-1/7*_alpha*(-3+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-2*_Z^2-_Z+1)).
a(n) = 5*a(n-2) - 6*a(n-4) + a(n-6). - Floor van Lamoen, Nov 02 2005
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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..40);
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MATHEMATICA
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LinearRecurrence[{1, 2, -1}, {1, 0, 2}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
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PROG
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(PARI) {a(n) = if(n==0, 1, if(n==1, 0, if(n==2, 2, a(n-1)+2*a(n-2)-a(n-3))))};
for(i=0, 40, print1(a(i), ", ")) \\ Lambert Klasen, Jan 30 2005
(MAGMA) I:=[1, 0, 2]; [n le 3 select I[n] else Self(n-1) + 2*Self(n-2) - Self(n-3): n in [1..40]]; // G. C. Greubel, May 08 2019
(Sage) ((1-x)/(1-x-2*x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 08 2019
(GAP) a:=[1, 0, 2];; for n in [4..40] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 08 2019
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CROSSREFS
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Cf. A096976; second differences of A028495 and first differences of A006053 (up to an offset).
Sequence in context: A074392 A284428 A096976 * A119245 A128731 A129157
Adjacent sequences: A052544 A052545 A052546 * 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, Jun 05 2000
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STATUS
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approved
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