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A277625
Nontrivial values of Fibonacci polynomials.
1
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
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).
Sequence in context: A085833 A066896 A369064 * A098177 A223577 A256793
KEYWORD
nonn
AUTHOR
Bobby Jacobs, Oct 24 2016
EXTENSIONS
More terms from Robert G. Wilson v, Oct 24 2016
STATUS
approved