A085151 -id:A085151 - cp's OEIS frontend

2, 3, 5, 8, 10, 12, 13, 17, 21, 26, 29, 33, 34, 37, 50, 55, 65, 70, 72, 82, 89, 101, 109, 122, 135, 144, 145, 169, 170, 197, 226, 228, 233, 257, 290, 305, 325, 357, 360, 362, 377, 401, 408, 442, 485, 528, 530, 577, 610, 626, 677, 701, 730, 747, 785, 842, 901, 962, 985, 987

Offset: 1

Bobby Jacobs, Oct 24 2016

nonn

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

			12 is in this sequence because FibonacciPolynomial(4, 2) = 12.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Wikipedia, Fibonacci polynomials.

Cf. A000045, A000129, A001076, A006190, A052918 (FibonacciPolynomial(x, y) for different values of y).
Cf. A002522, A054602, A085151 (FibonacciPolynomial(x, y) for different values of x).
Cf. A073133, A104244, A206296.

Take[Union[Flatten[Table[Fibonacci[x, y], {x, 3, 20}, {y, 50}]]], 60] (* Robert G. Wilson v, Oct 24 2016 *)

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

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

More terms from Robert G. Wilson v, Oct 24 2016

cp's OEIS Frontend

A277625 Nontrivial values of Fibonacci polynomials.

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