%I A078008
%S A078008 1,0,2,2,6,10,22,42,86,170,342,682,1366,2730,5462,10922,21846,43690,87382,
%T A078008 174762,349526,699050,1398102,2796202,5592406,11184810,22369622,44739242,
%U A078008 89478486,178956970,357913942,715827882,1431655766,2863311530,5726623062
%N A078008 Expansion of (1-x)/(1-x-2*x^2).
%C A078008 a(n) = A001045(n)+(-1)^n = A000079(n)-2*A001045(n). - Paul Barry (pbarry(AT)wit.ie), 
               Feb 20 2003
%C A078008 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
%C A078008 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
%C A078008 Permutations with one fixed point avoiding 123 and 132.
%C A078008 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
%C A078008 General form: iterate k=2^n-k. See also A001045. [From Vladimir Orlovsky 
               (4vladimir(AT)gmail.com), Dec 11 2008]
%C A078008 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), May 25 2009: 
               (Start)
%C A078008 Signed (A151575) [1, 0, 2, -2, 6, -10, 22, -42, 86,...] = an operator 
               for toothpick
%C A078008 sequences. The signed sequence (A151575) convolved with A151548 = toothpick 
               sequence
%C A078008 A139250. The signed sequence (A151575) convolved with A151555 = toothpick 
               sequence A153006. (End)
%D A078008 Paul Barry, A Catalan Transform and Related Transformations on Integer 
               Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
%D A078008 Leonhard Euler, Introductio in analysin infinitorum (1748), section 216.
%D A078008 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley 
               and Sons, N.Y., 1983, ex. 1.1.10a.
%H A078008 Beeler, M., Gosper, R. W., & Schroeppel, R. C., <a href="http://www.inwap.com/
               pdp10/hbaker/hakmem/number.html">R. HAKMEM. MIT AI Memo 239, Feb. 
               29, 1972</a>. (Item #54 by Salamin & Gosper)
%H A078008 T. Mansour and A. Robertson, <a href="http://arXiv.org/abs/math.CO/0204005">
               Refined restricted permutations...</a>.
%H A078008 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A078008 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A078008 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]
%F A078008 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
%F A078008 E.g.f. (exp(2x)+2exp(-x))/3. - Paul Barry (pbarry(AT)wit.ie), Apr 20 
               2003
%F A078008 a(n)=(1/3)(2^n+2(-1)^n) - Mario Catalani (mario.catalani(AT)unito.it), 
               Aug 29 2003
%F A078008 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
%F A078008 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
%F A078008 A137208(n+1)-2*A137208(n)=a(n) signed. - Paul Curtz (bpcrtz(AT)free.fr), 
               Aug 03 2008
%F A078008 a(n) = A001045(n+1)-A001045(n) - Paul Curtz (bpcrtz(AT)free.fr), Feb 
               09 2009
%t A078008 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]
%Y A078008 Cf. A001045 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Dec 11 
               2008]
%Y A078008 See A151575 for a signed version.
%Y A078008 Bisections: A047849, A020988. [From R. J. Mathar (mathar(AT)strw.leidenunvi.nl), 
               Feb 25 2009]
%Y A078008 Cf. A151548, A139250, A151555, A153006 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 25 2009]
%Y A078008 Sequence in context: A167399 A019310 A014113 this_sequence A151575 A076907 
               A153897
%Y A078008 Adjacent sequences: A078005 A078006 A078007 this_sequence A078009 A078010 
               A078011
%K A078008 nonn
%O A078008 0,3
%A A078008 N. J. A. Sloane (njas(AT)research.att.com), Nov 17 2002