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A078008
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Expansion of (1-x)/(1-x-2*x^2).
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101
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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; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) = A001045(n)+(-1)^n = A000079(n)-2*A001045(n). - Paul Barry (pbarry(AT)wit.ie), Feb 20 2003
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 (qntmpkt(AT)yahoo.com), 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 (pbarry(AT)wit.ie), Nov 17 2003
Permutations with one fixed point avoiding 123 and 132.
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 (reinhard.zumkeller(AT)gmail.com), Jun 10 2005
General form: iterate k=2^n-k. See also A001045. [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 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]
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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.
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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.
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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 (pbarry(AT)wit.ie), Feb 20 2003
E.g.f. (exp(2x)+2exp(-x))/3. - Paul Barry (pbarry(AT)wit.ie), Apr 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 (pbarry(AT)wit.ie), 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 (pbarry(AT)wit.ie), Jul 30 2004
A137208(n+1)-2*A137208(n)=a(n) signed. - Paul Curtz (bpcrtz(AT)free.fr), Aug 03 2008
a(n) = A001045(n+1)-A001045(n) - Paul Curtz (bpcrtz(AT)free.fr), 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 R. Janjic (agnus(AT)blic.net), 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]
A078008(n+1) = 2*A001045(n) [Benoît Jubin, Nov 22 2011]
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MATHEMATICA
| k=0; lst={1, k}; Do[k=2^n-k; AppendTo[lst, k], {n, 1, 5!}]; lst (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008 *)
CoefficientList[Series[(1-x)/(1-x-2x^2), {x, 0, 50}], x] (* From Harvey P. Dale, Mar 30 2011 *)
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PROG
| (PARI) a(n)=(1<<n+2*(-1)^n)/3 \\ Charles R Greathouse IV, Jun 10 2011
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CROSSREFS
| Cf. A001045 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 2008]
See A151575 for a signed version.
Bisections: A047849, A020988. [From R. J. Mathar (mathar(AT)strw.leidenunvi.nl), Feb 25 2009]
Cf. A151548, A139250, A151555, A153006 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009].
Sequence in context: A019310 A014113 * A151575 A076907 A153897 A103774
Adjacent sequences: A078005 A078006 A078007 * A078009 A078010 A078011
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002
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