A078008 Expansion of (1-x)/( (1+x)*(1-2*x) ).

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

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. 3*a(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

a(n) is the number of words of length n over a binary alphabet whose position in the lexicographic order is a multiple of three. a(3) = 2: aba, bab. - Alois P. Heinz, Apr 13 2022

a(n) is the number of words of length n over a ternary alphabet starting with a fixed letter (say, 'a') and ending in a different letter, such that no two adjacent letters are the same. a(4) = 6: abcab, abcac, abcbc, acbab, acbac, acbcb. - Ignat Soroko, Jul 19 2023

References

Kenneth Edwards, Michael A. Allen, A new combinatorial interpretation of the Fibonacci numbers squared, Part II, Fib. Q., 58:2 (2020), 169-177.

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, ex. 1.1.10a.

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000 (terms n=0..300 from T. D. Noe)

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.

Paul Barry, A Note on Riordan Arrays with Catalan Halves, arXiv:1912.01124 [math.CO], 2019.

Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.

Paul Barry and 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, and 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.

Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See p. 13.

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).

Index entries for sequences related to Chebyshev polynomials.

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 Lafont, Jun 01 2009]

a(n) = Sum_{k=0..floor(n/3)} binomial(n, f(n)+3*k) where f(n) = (0, 2, 1, 0, 2, 1, ...) = A080424(n). - Paul Barry, Feb 20 2003

E.g.f. (exp(2*x) + 2*exp(-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) = (2^n + 2*(-1)^n)/3. - Mario Catalani (mario.catalani(AT)unito.it), Aug 29 2003

a(n) = T(n, i/(2*sqrt(2)))(-i*sqrt(2)^n - U(n-1, i/(2*sqrt(2)))(-i*sqrt(2))^(n-1)/2 - Paul Barry, Nov 17 2003

From Paul Barry, Jul 30 2004: (Start)

a(n) = 2*a(n-1) + 2*(-1)^n for n > 0, with a(0)=1.

a(n) = Sum_{k=0..n} (-1)^k*(2^(n-k-1) + 0^(n-k)/2). (End)

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[1] =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

a(n) = (-1)^n * A151575(n). - G. C. Greubel, Jun 28 2019

a(n)+a(n+1) = 2^n. - R. J. Mathar, Feb 24 2021

a(n) = -a(2-n) * (-2)^(n-1) = (3/2)*(a(n-1) + a(n-2)) - a(n-3) for all n in Z. - Michael Somos, Mar 18 2022

Example

G.f. = 1 + 2*x^2 + 2*x^3 + 6*x^4 + 10*x^5 + 22*x^6 + ... - Michael Somos, Mar 18 2022

Mathematica

Table[(2^n + 2*(-1)^n)/3, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Dec 11 2008; modified by G. C. Greubel, Jun 28 2019 *)
CoefficientList[Series[(1-x)/(1-x-2x^2), {x, 0, 40}], x]  (* Harvey P. Dale, Mar 30 2011 *)
LinearRecurrence[{1, 2}, {1, 0}, 40] (* Jean-François Alcover, Sep 23 2017 *)

Prog

(PARI) a(n)=(1<<n+2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 10 2011
(Magma) [(2^n + 2*(-1)^n)/3: n in [0..40]]; // G. C. Greubel, Jun 28 2019
(Sage) [(2^n + 2*(-1)^n)/3 for n in (0..40)] # G. C. Greubel, Jun 28 2019
(GAP) List([0..40], n-> (2^n + 2*(-1)^n)/3) # G. C. Greubel, Jun 28 2019

Crossrefs

First differences of A001045.

See A151575 for a signed version.

Bisections: A047849, A020988.

Cf. A151548, A139250, A151555, A153006.

Sequence in context: A300274 A019310 A014113 * A151575 A284462 A262278

Adjacent sequences: A078005 A078006 A078007 * A078009 A078010 A078011

Keyword

nonn,easy

Author

N. J. A. Sloane, Nov 17 2002

Status

approved