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A047946 5*F(n)^2+3*(-1)^n where F(n) are the Fibonacci numbers A000045. 3
3, 2, 8, 17, 48, 122, 323, 842, 2208, 5777, 15128, 39602, 103683, 271442, 710648, 1860497, 4870848, 12752042, 33385283, 87403802, 228826128, 599074577, 1568397608, 4106118242, 10749957123, 28143753122, 73681302248 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Form the matrix A=[1,1,1;2,1,0;1,0,0]. a(n)=trace(A^n). - Paul Barry, Sep 22 2004

The set of prime divisors of elements of this sequence with the exception of 3 is the set of primes that do not divide odd Fibonacci numbers. - Tanya Khovanova, May 19 2008

LINKS

Table of n, a(n) for n=0..26.

Tanya Khovanova, Divisibility of Odd Fibonaccis

Index entries for linear recurrences with constant coefficients, signature (2,2,-1).

FORMULA

a(n)=F(3n)/F(n), n>0; a(n)=2*a(n-1)+2*a(n-2)-a(n-3); a(n)=3a(n-1)-a(n-2)+5(-1)^n; a(n) = L(2n) + (-1)^n, where the L(n) are Lucas numbers A000032.

G.f.: ( 3-4*x-2*x^2 ) / ( (1+x)*(x^2-3*x+1) ).

for n>0 a(n)=A000045(3n)/A000045(n) - Benoit Cloitre, Aug 30 2003

a(n)=[3/2+(1/2)*sqrt(5)]^n+(-1)^n+[3/2-(1/2)*sqrt(5)]^n, with n>=0 - Paolo P. Lava, Jun 12 2008

PROG

(PARI) a(n)=5*fibonacci(n)^2+3*(-1)^n

CROSSREFS

Cf. A000045, A000032.

Second row of array A028412.

Cf. A133247 = prime numbers p with property that no odd Fibonacci number is divisible by p.

Sequence in context: A171634 A107300 A285787 * A066045 A110866 A257958

Adjacent sequences:  A047943 A047944 A047945 * A047947 A047948 A047949

KEYWORD

nonn,easy

AUTHOR

John W. Layman, May 21 1999

EXTENSIONS

Entry improved by comments from Michael Somos.

STATUS

approved

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Last modified June 25 00:41 EDT 2017. Contains 288708 sequences.