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A214706
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a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=5.
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9
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1, 5, 5, 25, 125, 3125, 390625, 1220703125, 476837158203125, 582076609134674072265625, 277555756156289135105907917022705078125, 161558713389263217748322010169914619837072677910327911376953125
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OFFSET
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0,2
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COMMENTS
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a(17) has 1117 digits.
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 4*Sum_{n = 1..oo} 1/5^floor(n*phi) (= 16*Sum_{n = 1..oo} floor(n/phi)/5^n) = 0.83866 83869 91037 14262 ... = 1/(1 + 1/(5 + 1/(5 + 1/(25 + 1/(125 + 1/(3125 + 1/(390625 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/5^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)
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LINKS
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FORMULA
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a(n) = 5^Fibonacci(n).
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MAPLE
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a:= n-> 5^(<<1|1>, <1|0>>^n)[1, 2]:
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MATHEMATICA
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5^Fibonacci[Range[0, 11]]
nxt[{a_, b_}]:={b, a*b}; NestList[nxt, {1, 5}, 12][[All, 1]] (* Harvey P. Dale, Oct 14 2018 *)
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PROG
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(Magma) [5^Fibonacci(n): n in [0..13]];
(SageMath) [5^fibonacci(n) for n in range(15)] # G. C. Greubel, Jan 07 2024
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CROSSREFS
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Cf. A000045, A000301, A001622, A010098, A010099, A010100, A014565, A214706, A214887, A215270, A215271, A215272.
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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