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 A078008 Expansion of (1-x)/(1 - x - 2*x^2). 113
 1, 0, 2, 2, 6, 10, 22, 42, 86, 170, 342, 682, 1366, 2730, 5462, 10922, 21846, 43690, 87382, 174762, 349526, 699050, 1398102, 2796202, 5592406, 11184810, 22369622, 44739242, 89478486, 178956970, 357913942, 715827882, 1431655766, 2863311530, 5726623062 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Conjecture: a(n) = the number of fractions in the infinite Farey row of 2^n terms with even denominators. Compare the Salamin & Gosper item in the Beeler et al. link. - Gary W. Adamson, Oct 27 2003 Counts closed walks starting and ending at the same vertex of a triangle. 3a(n)=P(C_n,3) chromatic polynomial for 3 colors on cyclic graph C_n. A078008(n) + 2*A001045(n) = 2^n provides decomposition of Pascal's triangle. - Paul Barry, Nov 17 2003 Permutations with one fixed point avoiding 123 and 132. General form: iterate k -> 2^n-k. See also A001045. - Vladimir Joseph Stephan Orlovsky, Dec 11 2008 The inverse g.f. generates sequence 1, 0, -2, -2, -2, -2, ... a(n) gives the number of oriented (i.e., unreduced for symmetry) meanders on an (n+2) X 3 rectangular grid; see A201145. - Jon Wild, Nov 22 2011 Pisano period lengths: 1, 1, 6, 1, 4, 6, 6, 2, 18, 4, 10, 6, 12, 6, 12, 2, 8, 18, 18, 4, ... - R. J. Mathar, Aug 10 2012 a(n) is the number of length n binary words that end in an odd length run of 0's if we do not include the first letter of the word in our run length count. a(4) =6 because we have 0000, 0010, 0110, 1000, 1010, 1110. - Geoffrey Critzer, Dec 16 2013 a(n) is the top left entry of the n-th power of any of the six 3 X 3 matrices [0, 1, 1; 1, 1, 1; 1, 0, 0], [0, 1, 1; 1, 1, 0; 1, 1, 0], [0, 1, 1; 1, 0, 1; 1, 1, 0], [0, 1, 1; 1, 1, 0; 1, 0, 1], [0, 1, 1; 1, 0, 1; 1, 0, 1] or [0, 1, 1; 1, 0, 0; 1, 1, 1]. - R. J. Mathar, Feb 04 2014 a(n) is the number of compositions of n into parts of two kinds without part 1. - Gregory L. Simay, Jun 04 2018 REFERENCES I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 1.1.10a. LINKS T. D. Noe, Table of n, a(n) for n=0..300 David Applegate, Omar E. Pol and N. J. A. Sloane, The Toothpick Sequence and Other Sequences from Cellular Automata, Congressus Numerantium, Vol. 206 (2010), 157-191. [There is a typo in Theorem 6: (13) should read u(n) = 4.3^(wt(n-1)-1) for n >= 2.] Roland Bacher, Chebyshev polynomials, quadratic surds and a variation of Pascal's triangle, arXiv:1509.09054 [math.CO], 2015. See Section 4.6. Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5. Paul Barry, Jacobsthal Decompositions of Pascal's Triangle, Ternary Trees, and Alternating Sign Matrices, Journal of Integer Sequences, 19, 2016, #16.3.5. Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018. P. Barry, A. Hennessey, Notes on a Family of Riordan Arrays and Associated Integer Hankel Transforms , JIS 12 (2009) 09.5.3 M. Beeler, R. W. Gosper, R. C. Schroeppel, R. HAKMEM. MIT AI Memo 239, Feb 29 1972. (Item #54 by Salamin & Gosper) Ji Young Choi, A Generalization of Collatz Functions and Jacobsthal Numbers, J. Int. Seq., Vol. 21 (2018), Article 18.5.4. Leonhard Euler, Introductio in analysin infinitorum, (1748), section 216. T. Mansour and A. Robertson, Refined Restricted Permutations Avoiding Subsets of Patterns of Length Three, arXiv:math/0204005 [math.CO], 2002. N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS Index entries for linear recurrences with constant coefficients, signature (1,2). FORMULA Euler expands(1-x)/(1 - x - 2*x^2) into an infinite series and finds that the coefficient of the n-th term is (2^n + (-1)^n 2)/3. Section 226 shows that Euler could have easily found the recursion relation: a(n) = a(n-1) + 2a(n-2) with a(0) = 1 and a(1) = 0. - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006. [Typos corrected by Jaume Oliver i Lafont, Jun 01 2009] a(n) = sum_{k=0..floor(n, 3)} binomial(n, f(n)+3k) where f(n) = (0, 2, 1, 0, 2, 1, ...) = A080424(n). - Paul Barry, Feb 20 2003 E.g.f. (exp(2x) + 2exp(-x))/3. - Paul Barry, Apr 20 2003 a(n) = A001045(n) + (-1)^n = A000079(n) - 2*A001045(n). - Paul Barry, Feb 20 2003 a(n) = (1/3)(2^n + 2(-1)^n). - Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2003 a(n) = T(n, i/(2sqrt(2)))(-i*sqrt(2)^n - U(n-1, i/(2sqrt(2)))(-i*sqrt(2))^(n-1)/2 - Paul Barry, Nov 17 2003 a(0)=1, a(n) = 2a(n-1) + 2(-1)^n, n>0; a(n) = sum_{k=0..n} (-1)^k(2^(n-k-1) + 0^(n-k)/2). - Paul Barry, Jul 30 2004 a(n) = A014113(n-1) for n>0; a(n) = A052953(n-1) - 2*(n mod 2) = sum of n-th row of the triangle in A108561. - Reinhard Zumkeller, Jun 10 2005 A137208(n+1) - 2*A137208(n) = a(n) signed. - Paul Curtz, Aug 03 2008 a(n) = A001045(n+1) - A001045(n) - Paul Curtz, Feb 09 2009 If p =0, and p[i]=2, (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. - Milan Janjic, Apr 29 2010 a(n) = 2*(a(n - 2) + a(n - 3) + a(n - 4) .... + a(0)), that is, twice the sum of all the previous terms except the last; with a(0) = 1 and a(1) = 0. - Benoit Jubin, Nov 21 2011 a(n+1) = 2*A001045(n). - Benoit Jubin, Nov 22 2011 G.f.: 1 - x + x*Q(0), where Q(k) = 1 + 2*x^2 + (2*k+3)*x - x*(2*k+1 + 2*x)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013 G.f.: 1+ x^2*Q(0), where Q(k) = 1 + 1/(1 - x*(4*k+1+2*x)/(x*(4*k+3+2*x) + 1/Q(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 01 2014 a(n) = 3*a(n-2) + 2*a(n-3). - David Neil McGrath, Sep 10 2014 MATHEMATICA k=0; lst={1, k}; Do[k=2^n-k; AppendTo[lst, k], {n, 1, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008 *) CoefficientList[Series[(1-x)/(1-x-2x^2), {x, 0, 50}], x]  (* Harvey P. Dale, Mar 30 2011 *) LinearRecurrence[{1, 2}, {1, 0}, 35] (* Jean-François Alcover, Sep 23 2017 *) PROG (PARI) a(n)=(1<

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Last modified June 25 11:51 EDT 2019. Contains 324352 sequences. (Running on oeis4.)