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Search: id:A092054
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| A092054 |
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Logarithm base 2 of the sum of numerator and denominator of the convergents of the continued fraction expansion [1;1/2,1/3,1/4,...,1/n,...]. |
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+0
2
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| 1, 2, 4, 6, 7, 8, 11, 14, 15, 16, 18, 20, 21, 22, 26, 30, 31, 32, 34, 36, 37, 38, 41, 44, 45, 46, 48, 50, 51, 52, 57, 62, 63, 64, 66, 68, 69, 70, 73, 76, 77, 78, 80, 82, 83, 84, 88, 92, 93, 94, 96, 98, 99, 100, 103, 106, 107, 108, 110, 112, 113, 114, 120, 126, 127, 128, 130
(list; graph; listen)
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OFFSET
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1,2
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COMMENT
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Consider the convergents of the continued fraction expansion [1;1/2,1/3,1/4,...,1/n,...]. The numerators of the convergents are A001902 (successive denominators of Wallis's product approximation to Pi/2) and the denominators of the convergents are A092053. The sum of the numerators and the denominators equals a power of 2: A001902(n) + A092053(n) = 2^a(n).
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FORMULA
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2^a(n) = A001902(n) + A092053(n).
It appears that a(n) = sum_{k=1}^n A001511([(k+1)/2]). Equivalently, a(n) = 2n + 1 - A000120(n) - A000120(n+1) = A011371(n) + A011371(n+1). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Feb 02 2006
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EXAMPLE
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a(4)=6 since [1;1/2,1/3,1/4] = 1+1/(1/2+1/(1/3+1/(1/4))) = 45/19; and the sum of the numerator and denominator of 45/19 equals 45+19 = 2^6.
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PROGRAM
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(PARI) {a(n)=local(A); CF=contfracpnqn(vector(n, k, 1/k)); A=length(binary(numerator(1+CF[1, 1]/CF[2, 1])))-1}
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CROSSREFS
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Cf. A001902, A092053.
Sequence in context: A066507 A030493 A139527 this_sequence A029453 A014855 A096750
Adjacent sequences: A092051 A092052 A092053 this_sequence A092055 A092056 A092057
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Feb 19 2004
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