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A215272
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a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=9.
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9
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1, 9, 9, 81, 729, 59049, 43046721, 2541865828329, 109418989131512359209, 278128389443693511257285776231761, 30432527221704537086371993251530170531786747066637049
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OFFSET
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0,2
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COMMENTS
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Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 8*sum {n = 1..inf} 1/9^floor(n*phi) (= 64*sum {n = 1..inf} floor(n/phi)/9^n) = 0.90109 74122 99938 29901 ... = 1/(1 + 1/(9 + 1/(9 + 1/(81 + 1/(729 + 1/(59049 + 1/(43046721 + ...))))))). 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} 1/9^(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) = 9^Fibonacci(n).
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MAPLE
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a:= n-> 9^(<<1|1>, <1|0>>^n)[1, 2]:
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MATHEMATICA
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RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]
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PROG
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(Magma) [9^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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