

A051450


Number of positive rational knots with 2n+1 crossings.


9



1, 2, 5, 12, 30, 76, 195, 504, 1309, 3410, 8900, 23256, 60813, 159094, 416325, 1089648, 2852242, 7466468, 19546175, 51170460, 133962621, 350713222, 918170280, 2403786672, 6293172025, 16475700746, 43133883845, 112925875764
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

The number of positive rational knots with even crossing number is zero.
a(n) = (1/2)*(coefficient of x in the reduction by x^2>x+1 of the polynomial p(n,x) = 1+x^n+x^(2n)); see A192464. Reductions of polynomials by substitutions such as x^2>x+1 are introduced at A192232.  Clark Kimberling, Jul 01 2011


LINKS

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (4,3,2,1).


FORMULA

G.f. (version 1): x + (x/2)*(1/(1x/(4*(1x)^2) + x/(4*(1+x)^2)) + 1/(1x^2/(1x^4))).
G.f. (version 2): x*(12*x)/((1xx^2)*(13*x+x^2)).  N. J. A. Sloane, Jan 21 2001
Binomial transform of Fibonacci(n)*(1(1)^n)/2. Binomial transform of (Fibonacci(n) + Fibonacci(n))/2.  Paul Barry, 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))/(2*sqrt(5)) or a(n) = floor((1 + phi^n + (1+phi)^n)/(2*sqrt(5))).  Herbert Kociemba, May 12 2004
Also, number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and s(i)  s(i1) <= 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(2*Pi*k/5)*(1+2*cos(Pi*k/5))^n.  Herbert Kociemba, Jun 07 2004
a(n) = (Fibonacci(2*n) + Fibonacci(n))/2.  Vladeta Jovovic, Jul 17 2004
Convolution of F(n) and F(2n1). a(n) = Sum_{k=0..n} F(2k1)*F(nk).  Paul Barry, Jul 26 2004
a(n) = 4*a(n1)  3*a(n2)  2*a(n3) + a(n4).  Colin Barker, Nov 01 2014


EXAMPLE

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 AlexanderBriggs notation).


PROG

(PARI) Vec(x*(2*x1)/((x^23*x+1)*(x^2+x1)) + O(x^100)) \\ Colin Barker, Nov 01 2014
(MAGMA) [(Fibonacci(2*n)+Fibonacci(n))/2: n in [1..30]]; // Vincenzo Librandi, Nov 01 2014


CROSSREFS

Cf. A000045.
Sequence in context: A092247 A108360 A051163 * A038508 A105695 A244884
Adjacent sequences: A051447 A051448 A051449 * A051451 A051452 A051453


KEYWORD

easy,nonn


AUTHOR

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


EXTENSIONS

More terms from James A. Sellers, Dec 09 1999


STATUS

approved



