A051450 Number of positive rational knots with 2n+1 crossings.

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

Offset: 1

Author

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

Keywords

• easy
• nonn

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

Examples

			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).
		

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).

Crossrefs

Cf. A000045.

Programs

Magma

[(Fibonacci(2*n)+Fibonacci(n))/2: n in [1..30]]; // Vincenzo Librandi, Nov 01 2014

Maple

A051450 := proc(n)
    (combinat[fibonacci](n)+combinat[fibonacci](2*n))/2 ;
end proc:
seq(A051450(n),n=1..10) ; # R. J. Mathar, Nov 02 2021

Mathematica

LinearRecurrence[{4, -3, -2, 1}, {1, 2, 5, 12}, 30] (* Jean-François Alcover, Aug 28 2023 *)

PARI

Vec(x*(2*x-1)/((x^2-3*x+1)*(x^2+x-1)) + O(x^100)) \\ Colin Barker, Nov 01 2014

Formula

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)). - 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(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(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(2n-1). a(n) = Sum_{k=0..n} F(2k-1)*F(n-k). - Paul Barry, Jul 26 2004

a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4). - Colin Barker, Nov 01 2014

Extensions

More terms from James Sellers, Dec 09 1999