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A145232 a(n) = Fibonacci(5^n). 7
1, 5, 75025, 59425114757512643212875125, 18526362353047317310282957646406309593963452838196423660508102562977229905562196608078556292556795045922591488273554788881298750625 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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

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.

General (hyperbolic) trigonometric formula

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

a(n) = (2/sqrt(5))*cosh((2k+1)^n*arccosh(sqrt(5)/2)).

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..5

FORMULA

a(n) = (2/sqrt(5))*cosh(5^n*arccosh(sqrt(5)/2)).

a(n) = 5^n*A128935(n). - R. J. Mathar, Nov 04 2010

a(n) = A000045(A000351(n)). - Michel Marcus, Nov 07 2013

MATHEMATICA

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

Table[Round[N[(2/Sqrt[5])*Cosh[5^n*ArcCosh[Sqrt[5]/2]], 1000]], {n, 1, 4}]

Fibonacci[5^Range[0, 4]] (* Harvey P. Dale, Nov 29 2018 *)

CROSSREFS

Cf. A000045.

Cf. (k^n)-th Fibonacci number: A058635 (k=2), A045529 (k=3), A145231 (k=4), this sequence (k=5), A145233 (k=6), A145234 (k=7), A250487 (k=8), A250488 (k=9), A250489 (k=10).

Sequence in context: A247845 A050816 A171981 * A263174 A123591 A133381

Adjacent sequences:  A145229 A145230 A145231 * A145233 A145234 A145235

KEYWORD

nonn

AUTHOR

Artur Jasinski, Oct 05 2008

STATUS

approved

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Last modified November 28 19:48 EST 2021. Contains 349415 sequences. (Running on oeis4.)