A206296 - OEIS

A206296

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).

1, 2, 3, 10, 63, 2750, 842751, 85558343750, 2098355820117528699, 769999781728184386440152910156250, 2359414683424785920146467280333749864720543920418139851

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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.

LINKS

Table of n, a(n) for n=0..10.

Eric Weisstein's World of Mathematics, Fibonacci polynomial

Wikipedia, Fibonacci polynomials

FORMULA

From Antti Karttunen, Jul 29 2015: (Start)

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.]

A048675(a(n)) = A000129(n). [at x=2.]

A090880(a(n)) = A006190(n). [at x=3.]

(End)

EXAMPLE

n a(n) prime factorization Fibonacci polynomial

------------------------------------------------------------

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

MATHEMATICA

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 *)

PROG

(Scheme, with memoization-macro definec)

(definec (A206296 n) (cond ((<= n 1) (+ 1 n)) (else (* (A003961 (A206296 (- n 1))) (A206296 (- n 2))))))

(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])

print(l) # Indranil Ghosh, Jun 21 2017

CROSSREFS

Cf. A000045, A000129, A001222, A003961, A006190, A049310, A048675, A090880, A104244, A206284.

Other such mappings:

polynomial sequence integer sequence

-----------------------------------------

x^n A000040

(x+1)^n A007188

n*x^(n-1) A062457

(1-x^n)/(1-x) A002110

n + (n-1)x + ... +x^n A006939

Stern polynomials A260443

Sequence in context: A093856 A173097 A088221 * A124923 A291935 A088222

Adjacent sequences: A206293 A206294 A206295 * A206297 A206298 A206299

KEYWORD

nonn

AUTHOR

Clark Kimberling, Feb 05 2012

EXTENSIONS

a(0) = 1 prepended (to indicate 0-polynomial), Name changed, Comments and Example section rewritten by Antti Karttunen, Jul 29 2015

STATUS

approved