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%I #44 Aug 28 2023 08:21:58
%S 1,2,5,12,30,76,195,504,1309,3410,8900,23256,60813,159094,416325,
%T 1089648,2852242,7466468,19546175,51170460,133962621,350713222,
%U 918170280,2403786672,6293172025,16475700746,43133883845,112925875764
%N Number of positive rational knots with 2n+1 crossings.
%C The number of positive rational knots with even crossing number is zero.
%C 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
%H Colin Barker, <a href="/A051450/b051450.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-2,1).
%F 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))).
%F G.f. (version 2): x*(1-2*x)/((1-x-x^2)*(1-3*x+x^2)). - _N. J. A. Sloane_, Jan 21 2001
%F Binomial transform of Fibonacci(n)*(1-(-1)^n)/2. Binomial transform of (Fibonacci(n) + Fibonacci(-n))/2. - _Paul Barry_, Apr 23 2004
%F 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
%F 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
%F a(n) = (Fibonacci(2*n) + Fibonacci(n))/2. - _Vladeta Jovovic_, Jul 17 2004
%F 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
%F a(n) = 4*a(n-1) - 3*a(n-2) - 2*a(n-3) + a(n-4). - _Colin Barker_, Nov 01 2014
%e 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).
%p A051450 := proc(n)
%p (combinat[fibonacci](n)+combinat[fibonacci](2*n))/2 ;
%p end proc:
%p seq(A051450(n),n=1..10) ; # _R. J. Mathar_, Nov 02 2021
%t LinearRecurrence[{4, -3, -2, 1}, {1, 2, 5, 12}, 30] (* _Jean-François Alcover_, Aug 28 2023 *)
%o (PARI) Vec(x*(2*x-1)/((x^2-3*x+1)*(x^2+x-1)) + O(x^100)) \\ _Colin Barker_, Nov 01 2014
%o (Magma) [(Fibonacci(2*n)+Fibonacci(n))/2: n in [1..30]]; // _Vincenzo Librandi_, Nov 01 2014
%Y Cf. A000045.
%K easy,nonn
%O 1,2
%A Alexander Stoimenow (stoimeno(AT)math.toronto.edu)
%E More terms from _James A. Sellers_, Dec 09 1999
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