1,2

The number of positive rational knots with even crossing number is zero.

G.f. (version 1): -x+x/2*(1/(1-x/(4*(1-x)^2)+x/(4*(1+x)^2))+1/(1-x^2/(1-x^4))).

G.f. (version 2): x*(1-2*x)/((1-x-x^2)*(1-3*x+x^2)) - njas, Jan 21, 2001

Binomial transform of Fib(n)(1-(-1)^n)/2. Binomial transform of (Fib(n)+Fib(-n))/2. - Paul Barry (pbarry(AT)wit.ie), Apr 23 2004

Let phi be the golden ratio (1+Sqrt[5])/2. Then a(n)= (phi^n - (-phi)^(-n) + (1+phi)^n - (1+phi)^(-n))/(2Sqrt[5]) or a(n) = Floor[(1 + phi^n + (1+phi)^n)/(2Sqrt[5])] - Herbert Kociemba (kociemba(AT)t-online.de), May 12 2004

Also, number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1, 2, ...., n, s(0) = 1, s(n) = 2. a(n)=(2/5)*Sum(k, 1, 4, Sin(Pi*k/5)Sin(2Pi*k/5)(1+2Cos(Pi*k/5))^n) - Herbert Kociemba (kociemba(AT)t-online.de), Jun 07 2004

a(n) = (Fibonacci(2*n)+Fibonacci(n))/2. - Vladeta Jovovic (vladeta(AT)Eunet.yu), Jul 17 2004

Convolution of F(n) and F(2n-1). a(n)=sum{k=0..n, F(2k-1)F(n-k)} - Paul Barry (pbarry(AT)wit.ie), Jul 26 2004

a(4)=12 because we have 12 positive rational knots with 9 crossings: 9_1 to 9_7, 9_9, 9_10, 9_13, 9_18 and 9_23 (in Alexander-Briggs notation)

Cf. A000045.

Sequence in context: A092247 A108360 A051163 this_sequence A038508 A002026 A105695

Adjacent sequences: A051447 A051448 A051449 this_sequence A051451 A051452 A051453

easy,nonn

Alexander Stoimenow (stoimeno(AT)math.toronto.edu)

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 09 1999