A277625 Nontrivial values of Fibonacci polynomials.

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

1,1

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

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.

Sequence in context: A085833 A066896 A369064 * A098177 A223577 A256793

Adjacent sequences: A277622 A277623 A277624 * A277626 A277627 A277628

Keyword

nonn

Author

Bobby Jacobs, Oct 24 2016

Extensions

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

Status

approved