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A071766
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Denominator of the continued fraction expansion whose terms are the first-order differences of exponents in the binary representation of 4*n, with the exponents of 2 being listed in descending order.
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6
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1, 1, 1, 2, 1, 2, 3, 3, 1, 2, 3, 3, 4, 5, 4, 5, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13, 6, 9, 10, 11, 9, 12, 11, 13, 1, 2, 3, 3, 4, 5, 4, 5, 5, 7, 7, 8, 5, 7, 7, 8, 6, 9, 10, 11, 9, 12, 11, 13
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OFFSET
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0,4
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LINKS
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Paul D. Hanna, Table of n, a(n) for n = 0..10000
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FORMULA
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a(n) = A071585(m), where m = n - floor(log(n)/log(2));
a(0) = 1, a(2^k) = 1, a(2^k + 1) = 2.
a(2^k - 1) = Fibonacci(k+1) = A000045(k+1).
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EXAMPLE
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a(37)=5 as it is the denominator of 17/5 = 3 + 1/(2 + 1/2), which is a continued fraction that can be derived from the binary expansion of 4*37 = 2^7 + 2^4 + 2^2; the binary exponents are {7, 4, 2}, thus the differences of these exponents are {3, 2, 2}; giving the continued fraction expansion of 17/5=[3,2,2].
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PROG
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(PARI) {a(n)=local(N=4*n, E=#binary(N)-1, P=[E], CF); while(E>1, P=concat(P, E=#binary(N=N-2^E)-1)); CF=Vec(Ser(P)*(x-1)); if(n==0, CF[1]=1, CF[1]=0); contfracpnqn(CF)[1, 1]} \\ Paul D. Hanna, Feb 22 2013
for(n=0, 256, print1(a(n), ", "))
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CROSSREFS
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Cf. A071585.
Sequence in context: A033803 A035531 A118977 * A007305 A112531 A100002
Adjacent sequences: A071763 A071764 A071765 * A071767 A071768 A071769
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KEYWORD
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easy,nonn,frac
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AUTHOR
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Paul D. Hanna, Jun 04 2002
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STATUS
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approved
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