|
%I #53 Nov 04 2016 05:24:36
%S 2,3,5,8,10,12,13,17,21,26,29,33,34,37,50,55,65,70,72,82,89,101,109,
%T 122,135,144,145,169,170,197,226,228,233,257,290,305,325,357,360,362,
%U 377,401,408,442,485,528,530,577,610,626,677,701,730,747,785,842,901,962,985,987
%N Nontrivial values of Fibonacci polynomials.
%C The polynomial FibonacciPolynomial(x, y) satisfies the recurrence FibonacciPolynomial(0, y) = 0, FibonacciPolynomial(1, y) = 1, and FibonacciPolynomial(x, y) = y*FibonacciPolynomial(x-1, y) + FibonacciPolynomial(x-2, y).
%C Nontrivial means a value FibonacciPolynomial(x, y) with x>=3 and y>=1. For FibonacciPolynomial(0, y) = 0 and FibonacciPolynomial(1, y) = 1 for all y, and any number y can be represented trivially as FibonacciPolynomial(2, y).
%C 5 = FibonacciPolynomial(5, 1) = FibonacciPolynomial(3, 2) is the only known number that can be represented as a nontrivial Fibonacci polynomial in more than one way.
%C Numbers obtained as A104244(n,A206296(k)), where n >= 1 and k >= 3 (all terms from array A073133 except its two leftmost columns) and then sorted into ascending order, with any possible duplicate (5) removed. - _Antti Karttunen_, Oct 29 2016
%H Charles R Greathouse IV, <a href="/A277625/b277625.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Fibonacci_polynomials">Fibonacci polynomials</a>.
%F FibonacciPolynomial(x, y) with x>=3 and y>=1.
%F a(n) = n^2 - 2*n^(5/3) - O(n^(3/2)). - _Charles R Greathouse IV_, Nov 03 2016
%e 12 is in this sequence because FibonacciPolynomial(4, 2) = 12.
%t Take[Union[Flatten[Table[Fibonacci[x, y], {x, 3, 20}, {y, 50}]]], 60] (* _Robert G. Wilson v_, Oct 24 2016 *)
%o (PARI) list(lim)=my(v=List()); for(y=1,sqrtint(lim\1-1), my(a=y,b=y^2+1); while(b<=lim, listput(v,b); [a,b]=[b,a+y*b])); Set(v) \\ _Charles R Greathouse IV_, Oct 30 2016
%Y Cf. A000045, A000129, A001076, A006190, A052918 (FibonacciPolynomial(x, y) for different values of y).
%Y Cf. A002522, A054602, A085151 (FibonacciPolynomial(x, y) for different values of x).
%Y Cf. A073133, A104244, A206296.
%K nonn
%O 1,1
%A _Bobby Jacobs_, Oct 24 2016
%E More terms from _Robert G. Wilson v_, Oct 24 2016
|
