0,3

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. - 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)+2A001045(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)lhsystems.com), Jun 10 2005

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-1) with a(1)= 1 and a(2) = 0. - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006

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

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.

T. Mansour and A. Robertson, Refined restricted permutations....

Index entries for sequences related to Chebyshev polynomials.

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

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)))(-isqrt(2)^n-U(n-1, i/(2sqrt(2)))(-isqrt(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

Expansion of (1-x)/(1-x-2*x^2). - V. Frederick Rickey (fred-rickey(AT)usma.edu), Feb 10 2006

Adjacent sequences: A078005 A078006 A078007 this_sequence A078009 A078010 A078011

Sequence in context: A123757 A019310 A014113 this_sequence A076907 A103774 A036052

nonn

njas, Nov 17 2002