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A052547 Expansion of (1-x)/(1-x-2*x^2+x^3).

%I A052547 #64 Jul 02 2025 16:01:58
%S A052547 1,0,2,1,5,5,14,19,42,66,131,221,417,728,1341,2380,4334,7753,14041,
%T A052547 25213,45542,81927,147798,266110,479779,864201,1557649,2806272,
%U A052547 5057369,9112264,16420730,29587889,53317085,96072133,173118414,311945595,562110290,1012883066
%N A052547 Expansion of (1-x)/(1-x-2*x^2+x^3).
%C A052547 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
%C A052547 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
%C A052547 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.
%C A052547 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
%H A052547 G. C. Greubel, <a href="/A052547/b052547.txt">Table of n, a(n) for n = 0..1000</a>
%H A052547 Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H A052547 Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, <a href="https://arxiv.org/abs/2210.12411">On a variant of Flory model</a>, arXiv:2210.12411 [math.CO], 2022.
%H A052547 Man Gao, Colin Tan, and Jie Wu, <a href="https://doi.org/10.1016/j.topol.2016.06.015">Loop homological invariants associated to real projective spaces</a>, Topology Appl. 209, 275-288 (2016).
%H A052547 M. Gao and J. Wu, <a href="http://arxiv.org/abs/1301.0864">Homology Decompositions of the Loops on 1-Stunted Borel Constructions of C_2-Actions</a>, arXiv:1301.0864 [math.AT], 2013.
%H A052547 Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 15.
%H A052547 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=483">Encyclopedia of Combinatorial Structures 483</a>
%H A052547 P. Steinbach, <a href="http://www.jstor.org/stable/2691048">Golden fields: a case for the heptagon</a>, Math. Mag. 70 (1997), no. 1, 22-31.
%H A052547 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1).
%F A052547 a(n) = a(n-1) + 2*a(n-2) - a(n-3), with a(0)=1, a(1)=0, a(2)=2.
%F A052547 a(n) = Sum(-1/7*_alpha*(-3+_alpha)*_alpha^(-1-n), _alpha=RootOf(_Z^3-2*_Z^2-_Z+1)).
%F A052547 a(n) = 5*a(n-2) - 6*a(n-4) + a(n-6). - _Floor van Lamoen_, Nov 02 2005
%p A052547 spec := [S,{S=Sequence(Prod(Z,Union(Z,Prod(Z, Sequence(Z)))))},unlabeled]: seq(combstruct[count](spec,size=n), n=0..40);
%t A052547 LinearRecurrence[{1, 2, -1}, {1, 0, 2}, 40] (* _Vladimir Joseph Stephan Orlovsky_, Feb 13 2012 *)
%o A052547 (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))))};
%o A052547 for(i=0,40,print1(a(i),",")) \\ Lambert Klasen, Jan 30 2005
%o A052547 (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
%o A052547 (Sage) ((1-x)/(1-x-2*x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 08 2019
%o A052547 (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
%Y A052547 Cf. A096976; second differences of A028495 and first differences of A006053 (up to an offset).
%K A052547 easy,nonn
%O A052547 0,3
%A A052547 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052547 More terms from _James Sellers_, Jun 05 2000