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A206296
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Prime factorization representation of Fibonacci polynomials: a(0) = 1, a(1) = 2, and for n > 1, a(n) = A003961(a(n-1)) * a(n-2).
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20
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1, 2, 3, 10, 63, 2750, 842751, 85558343750, 2098355820117528699, 769999781728184386440152910156250, 2359414683424785920146467280333749864720543920418139851
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OFFSET
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0,2
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COMMENTS
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These are numbers matched to the Fibonacci polynomials according to the scheme explained in A206284 (see also A104244). In this case, the exponent of the k-th prime p_k in the prime factorization of a(n) indicates the coefficient of term x^(k-1) in the n-th Fibonacci polynomial. See the examples.
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LINKS
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FORMULA
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a(0) = 1, a(1) = 2, and for n >= 2, a(n) = A003961(a(n-1)) * a(n-2).
Other identities. For all n >= 0:
A001222(a(n)) = A000045(n). [When each polynomial is evaluated at x=1.]
(End)
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EXAMPLE
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n a(n) prime factorization Fibonacci polynomial
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0 1 (empty) F_0(x) = 0
1 2 p_1 F_1(x) = 1
2 3 p_2 F_2(x) = x
3 10 p_3 * p_1 F_3(x) = x^2 + 1
4 63 p_4 * p_2^2 F_4(x) = x^3 + 2x
5 2750 p_5 * p_3^3 * p_1 F_5(x) = x^4 + 3x^2 + 1
6 842751 p_6 * p_4^4 * p_2^3 F_6(x) = x^5 + 4x^3 + 3x
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MATHEMATICA
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c[n_] := CoefficientList[Fibonacci[n, x], x]
f[n_] := Product[Prime[k]^c[n][[k]], {k, 1, Length[c[n]]}]
Table[f[n], {n, 1, 11}] (* A206296 *)
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PROG
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(Scheme, with memoization-macro definec)
(Python)
from sympy import factorint, prime, primepi
from operator import mul
def a003961(n):
F=factorint(n)
return 1 if n==1 else reduce(mul, [prime(primepi(i) + 1)**F[i] for i in F])
l=[1, 2]
for n in range(2, 11):
l.append(a003961(l[n - 1])*l[n - 2])
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CROSSREFS
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Other such mappings:
polynomial sequence integer sequence
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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a(0) = 1 prepended (to indicate 0-polynomial), Name changed, Comments and Example section rewritten by Antti Karttunen, Jul 29 2015
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STATUS
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approved
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