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A277625
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Nontrivial values of Fibonacci polynomials.
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1
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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FibonacciPolynomial(x, y) with x>=3 and y>=1.
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EXAMPLE
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12 is in this sequence because FibonacciPolynomial(4, 2) = 12.
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MATHEMATICA
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Take[Union[Flatten[Table[Fibonacci[x, y], {x, 3, 20}, {y, 50}]]], 60] (* Robert G. Wilson v, Oct 24 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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