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A214887
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a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=7.
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9
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1, 7, 7, 49, 343, 16807, 5764801, 96889010407, 558545864083284007, 54116956037952111668959660849, 30226801971775055948247051683954096612865741943
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OFFSET
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0,2
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COMMENTS
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a(17) has 1350 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 := 6*sum {n = 1..inf} 1/7^floor(n*phi) (= 36*sum {n = 1..inf} floor(n/phi)/7^n) = 0.87718 67194 00499 51922 ... = 1/(1 + 1/(7 + 1/(7 + 1/(49 + 1/(343 + 1/(16807 + 1/(5764801 + ...))))))). 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} 1/7^(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) = 7^Fibonacci(n).
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MAPLE
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a:= n-> 7^(<<1|1>, <1|0>>^n)[1, 2]:
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MATHEMATICA
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7^Fibonacci[Range[0, 10]]
nxt[{a_, b_}]:={b, a*b}; Transpose[NestList[nxt, {1, 7}, 10]][[1]] (* Harvey P. Dale, Jun 10 2014 *)
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PROG
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(Magma) [7^Fibonacci(n): n in [0..10]];
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CROSSREFS
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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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