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 A052534 Expansion of (1-x)*(1+x)/(1-2*x-x^2+x^3). 11
 1, 2, 4, 9, 20, 45, 101, 227, 510, 1146, 2575, 5786, 13001, 29213, 65641, 147494, 331416, 744685, 1673292, 3759853, 8448313, 18983187, 42654834, 95844542, 215360731, 483911170, 1087338529, 2443227497, 5489882353, 12335653674 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Pairwise sums of A006356. Cf. A033303, A077850. - Ralf Stephan, Jul 06 2003 Number of (3412, P)-avoiding involutions in S_{n+1}, where P={1342, 1423, 2314, 3142, 2431, 4132, 3241, 4213, 21543, 32154, 43215, 15432, 53241, 52431, 42315, 15342, 54321}. - Ralf Stephan, Jul 06 2003 Number of 31- and 22-avoiding words of length n on alphabet {1,2,3} which do not end in 3 (e.g., at n=3, we have 111, 112, 121, 132, 211, 212, 232, 321 and 332). See A028859, A001519. - Jon Perry, Aug 04 2003 Form the graph with matrix A=[1, 1, 1; 1, 0, 0; 1, 0, 1]. Then the sequence 1,1,2,4,... with g.f. (1-x-x^2)/(1-2x-x^2+x^3) counts closed walks of length n at the degree 3 vertex. - Paul Barry, Oct 02 2004 a(n) is the number of Motzkin (n+1)-sequences whose flatsteps all occur at level <=1 and whose height is <=2. For example, a(5)=45 counts all 51 Motzkin 6-paths except FUUFDD, UFUFDD, UUFDDF, UUFDFD, UUFFDD, UUUDDD (the first five violate the flatstep restriction and the last violates the height restriction). - David Callan, Dec 09 2004 From Paul Barry, Nov 03 2010: (Start) The g.f. of 1,1,2,4,9,... can be expressed as 1/(1-x/(1-x/(1-x^2)) and as 1/(1-x-x^2/(1-x-x^2)). The second expression shows the link to the Motzkin numbers. (End) From Emeric Deutsch, Oct 31 2010: (Start) a(n) is the number of compositions of n into odd summands when we have two kinds of 1's. Proof: the g.f. of the set S={1,1',3,5,7,...} is g=2x+x^3/(1-x^2) and the g.f. of finite sequences of elements of S is 1/(1-g). Example: a(4)=20 because we have 1+3, 1'+3, 3+1, 3+1', and 2^4=16 of sums x+y+z+u, where x,y,z,u are taken from {1,1'}. (End) a(n-1) is the top left entry of the n-th power of any of the six 3 X 3 matrices [1, 1, 0; 1, 1, 1; 0, 1, 0] or [1, 1, 1; 0, 1, 1; 1, 1, 0] or [1, 0, 1; 1, 1, 1; 1, 1, 0] or [1, 1, 1; 1, 0, 1; 0, 1, 1] or [1, 0, 1; 0, 0, 1; 1, 1, 1] or [1, 1, 0; 1, 0, 1; 1, 1, 1]. - R. J. Mathar, Feb 03 2014 LINKS G. C. Greubel, Table of n, a(n) for n = 0..1000 Jean-Luc Baril, Rigoberto Flórez, and José L. Ramírez, Counting symmetric and asymmetric peaks in motzkin paths with air pockets, Univ. Bourgogne (France, 2023). C. P. de Andrade, J. P. de Oliveira Santos, E. V. P. da Silva and K. C. P. Silva, Polynomial Generalizations and Combinatorial Interpretations for Sequences Including the Fibonacci and Pell Numbers, Open Journal of Discrete Mathematics, 2013, 3, 25-32 doi:10.4236/ojdm.2013.31006. - From N. J. A. Sloane, Feb 20 2013 E. S. Egge, Restricted 3412-Avoiding Involutions: Continued Fractions, Chebyshev Polynomials and Enumerations, arXiv:math/0307050 [math.CO], 2003, sec. 8. Rigoberto Flórez and José L. Ramírez, Some enumerations on non-decreasing Motzkin paths, Australasian J. of Combinatorics (2018) Vol. 72(1), 138-154. Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 15. INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 464 S. Morier-Genoud, V. Ovsienko, and S. Tabachnikov, Introducing supersymmetric frieze patterns and linear difference operators, Math. Z. 281 (2015) 1061. Alexey Ustinov, Supercontinuants, arXiv:1503.04497 [math.NT], 2015. R. Witula, D. Slota and A. Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3. Index entries for linear recurrences with constant coefficients, signature (2,1,-1). FORMULA G.f.: (1 - x^2)/(1 - 2*x - x^2 + x^3). a(n) = 2*a(n-1) + a(n-2) - a(n-3), with a(0)=1, a(1)=2, a(2)=4. a(n) = Sum_{alpha = RootOf(1-2*x-x^2+x^3)} (1/7)*(2 + alpha)*alpha^(-1-n). a(n) = central term in the (n+1)-th power of the 3 X 3 matrix (shown in the example of A066170): [1 1 1 / 1 1 0 / 1 0 0]. E.g. a(6) = 101 since the central term in M^7 = 101. - Gary W. Adamson, Feb 01 2004 a(n) = A006054(n+2) - A006054(n). - Vladimir Kruchinin, Sep 09 2010 a(n) = A077998(n+2) - 2*A006054(n+2), which implies 7*a(n-2) = (2 + c(4) - 2*c(2))*(1 + c(1))^n + (2 + c(1) - 2*c(4))*(1 + c(2))^n + (2 + c(2) - 2*c(1))*(1 + c(4))^n, where c(j)=2*Cos(2Pi*j/7), a(-2)=a(-1)=1 since A077998 and A006054 are equal to the respective quasi-Fibonacci numbers. [Witula, Slota and Warzynski] - Roman Witula, Aug 07 2012 a(n+1) = A033303(n+1) - A033303(n). - Roman Witula, Sep 14 2012 a(n) = A006054(n+2)-A006054(n). - R. J. Mathar, Nov 23 2020 a(n) = A028495(-1-n) for all n in Z. - Michael Somos, Dec 12 2023 EXAMPLE G.f. = 1 + 2*x + 4*x^2 + 9*x^3 + 20*x^4 + 45*x^5 + 101*x^6 + 227*x^7 + 510*x^8 + ... - Michael Somos, Dec 12 2023 MAPLE spec := [S, {S=Sequence(Union(Z, Prod(Z, Sequence(Prod(Z, Z)))))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); MATHEMATICA LinearRecurrence[{2, 1, -1}, {1, 2, 4}, 40] (* Roman Witula, Aug 07 2012 *) CoefficientList[Series[(1-x^2)/(1-2x-x^2+x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Mar 17 2015 *) a[ n_] := {0, 1, 0} . MatrixPower[{{1, 1, 1}, {1, 1, 0}, {1, 0, 0}}, n+1] . {0, 1, 0}; (* Michael Somos, Dec 12 2023 *) PROG (Maxima) h(n):=if n=0 then 1 else sum(sum(binomial(k, j)*binomial(j, n-3*k+2*j)*2^(3*k-n-j)*(-1)^(k-j), j, 0, k), k, 1, n); a(n):=if n<2 then h(n) else h(n)-h(n-2); /* Vladimir Kruchinin, Sep 09 2010 */ (Magma) [n le 3 select 2^(n-1) else 2*Self(n-1)+Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Mar 17 2015 (PARI) my(x='x+O('x^40)); Vec((1-x^2)/(1-2*x-x^2+x^3)) \\ G. C. Greubel, May 09 2019 (PARI) {a(n) = [0, 1, 0] * [1, 1, 1; 1, 1, 0; 1, 0, 0]^(n+1) * [0, 1, 0]~}; /* Michael Somos, Dec 12 2023 */ (SageMath) ((1-x^2)/(1-2*x-x^2+x^3)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, May 09 2019 (GAP) a:=[1, 2, 4];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, May 09 2019 CROSSREFS Cf. A028495, A066170. Sequence in context: A085584 A369614 A080019 * A213411 A080135 A227978 Adjacent sequences: A052531 A052532 A052533 * A052535 A052536 A052537 KEYWORD nonn,easy AUTHOR encyclopedia(AT)pommard.inria.fr, Jan 25 2000 STATUS approved

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Last modified August 13 22:54 EDT 2024. Contains 375146 sequences. (Running on oeis4.)