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A145231 a(n) = Fibonacci(4^n). 7
1, 3, 987, 10610209857723, 141693817714056513234709965875411919657707794958199867 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Family of sequences (G^(k^n) - (1 - G)^(k^n))/sqrt(5)

k=2 see A058635

k=3 see A045529

k=4 see A145231

k=5 see A145232

k=6 see A145233

k=7 see A145234

This sequence has the property that a(n+1) is divisible by a(n). Conjecture: each prime divisor can occur only once (i.e. all terms are squarefree). - Artur Jasinski, Oct 05 2008

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..6

FORMULA

a(n) = (G^(4^n) - (1-G)^(4^n) )/sqrt(5) where G = (1 + sqrt 5)/2 = A001622.

a(n) = round( sqrt(4/5) *cosh( 4^n*arccosh (sqrt(5/4)) )).

a(n)= A000045(A000302(n)). - Michel Marcus, Nov 07 2013

MATHEMATICA

G = (1 + Sqrt[5])/2; Table[Expand[(G^(4^n) - (1 - G)^(4^n))/Sqrt[5]], {n, 1, 6}]

Table[Round[(4/5)^(1/2)*Cosh[4^n*ArcCosh[((5/4)^(1/2))]]], {n, 1, 7}]

Fibonacci[4^Range[5]] (* Harvey P. Dale, Mar 28 2012 *)

CROSSREFS

Cf. A000045.

Cf. (k^n)-th Fibonacci number: A058635 (k=2), A045529 (k=3), this sequence (k=4), A145232 (k=5), A145233 (k=6), A145234 (k=7), A250487 (k=8), A250488 (k=9), A250489 (k=10).

Sequence in context: A151585 A286525 A030250 * A318480 A167069 A024046

Adjacent sequences:  A145228 A145229 A145230 * A145232 A145233 A145234

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 05 2008

STATUS

approved

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Last modified December 1 12:38 EST 2021. Contains 349429 sequences. (Running on oeis4.)