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A078008 Expansion of (1-x)/(1-x-2*x^2). 104
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. [From 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)X3 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 3X3 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

REFERENCES

Leonhard Euler, Introductio in analysin infinitorum (1748), section 216.

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

Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.

Beeler, M., Gosper, R. W., & Schroeppel, R. C., R. HAKMEM. MIT AI Memo 239, Feb 29 1972. (Item #54 by Salamin & Gosper)

T. Mansour and A. Robertson, Refined restricted permutations..., arXiv:math.CO/0204005

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to 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 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[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. [From 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 [Benoît Jubin, Nov 21 2011]

a(n+1) = 2*A001045(n) [Benoît 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

MATHEMATICA

k=0; lst={1, k}; Do[k=2^n-k; AppendTo[lst, k], {n, 1, 5!}]; lst (* From 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 *)

PROG

(PARI) a(n)=(1<<n+2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 10 2011

CROSSREFS

Cf. A001045 [From Vladimir Joseph Stephan Orlovsky, Dec 11 2008]

See A151575 for a signed version.

Bisections: A047849, A020988. [From R. J. Mathar, Feb 25 2009]

Cf. A151548, A139250, A151555, A153006 [From Gary W. Adamson, May 25 2009].

Sequence in context: A019310 A014113 * A151575 A208900 A229733 A076907

Adjacent sequences:  A078005 A078006 A078007 * A078009 A078010 A078011

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Nov 17 2002

STATUS

approved

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Last modified April 23 19:18 EDT 2014. Contains 240946 sequences.