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A215270
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a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=6.
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9
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1, 6, 6, 36, 216, 7776, 1679616, 13060694016, 21936950640377856, 286511799958070431838109696, 6285195213566005335561053533150026217291776, 1800782593726645086383198950649858141454002621435149880441896326660096
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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 := 5*sum {n = 1..inf} 1/6^floor(n*phi) (= 25*sum {n = 1..inf} floor(n/phi)/6^n) = 0.86045 01626 86090 61353 ... = 1/(1 + 1/(6 + 1/(6 + 1/(36 + 1/(216 + 1/(7776 + 1/(1679616 + ...))))))). 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} 1/6^(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) = 6^Fibonacci(n).
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MAPLE
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a:= n-> 6^(<<1|1>, <1|0>>^n)[1, 2]:
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MATHEMATICA
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RecurrenceTable[{a[0] == 1, a[1] == 6, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]
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PROG
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(Magma) [6^Fibonacci(n): n in [0..11]];
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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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