OFFSET
1,1
COMMENTS
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).
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).
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.
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
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Wikipedia, Fibonacci polynomials.
FORMULA
FibonacciPolynomial(x, y) with x>=3 and y>=1.
a(n) = n^2 - 2*n^(5/3) - O(n^(3/2)). - Charles R Greathouse IV, Nov 03 2016
EXAMPLE
12 is in this sequence because FibonacciPolynomial(4, 2) = 12.
MATHEMATICA
Take[Union[Flatten[Table[Fibonacci[x, y], {x, 3, 20}, {y, 50}]]], 60] (* Robert G. Wilson v, Oct 24 2016 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Bobby Jacobs, Oct 24 2016
EXTENSIONS
More terms from Robert G. Wilson v, Oct 24 2016
STATUS
approved