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A066212
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A measure of how close the golden ratio is to rational numbers.
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3
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-3, 8, -21, 8, 55, -21, 21, -144, -21, 55, -55, 29, 377, -40, 55, -144, -34, 144, -74, 55, -987, -55, 107, -144, 55, 377, -86, 92, -377, -65, 195, -144, 84, 2584, -95, 144, -279, 78, 377, -144, 121, -987, -101, 227, -239, 107, 987, -144, 173, -509, -106, 377, -217, 144, -6765, -144, 250, -377, 127, 731, -203
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| New peaks (in absolute terms) occur when n is a Fibonacci number (1,2,3,5,8,13,21, ... A000045) and take alternate Fibonacci values with alternating signs (-3, 8, -21, 55, -144, 377, -987 ... A001906). Each new peak (after the second) appears twice (with different signs) between the next two peaks, when n is twice a Fibonacci number (4, 6, 10, 16, 26 ... and when n is a Lucas number (4, 7, 11, 18, 29 ... A000204); the larger peaks appear many more times later in the sequence.
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LINKS
| Harry J. Smith, Table of n, a(n) for n=1,...,1000
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FORMULA
| a(n) = round[1/(phi-round[phi*n]/n)] where phi = (sqrt(5)+1)/2 = 1.6180339... = round[1/(phi-A007067(n)/n)]
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EXAMPLE
| a(5) = round[1/(1.6180339-round[1.6180339*5]/5)] = round[1/(1.6180339-8/5)] = round[55.45] = 55, i.e. 8/5 is the closest multiple of 1/5 to phi and phi-8/5 is about 1/55.
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PROG
| (PARI) { phi=(1 + sqrt(5))/2; for (n=1, 1000, a=round(1/(phi - round(phi*n)/n)); write("b066212.txt", n, " ", a) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Feb 06 2010]
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CROSSREFS
| Sequence in context: A110861 A027320 A027319 * A171437 A075719 A101643
Adjacent sequences: A066209 A066210 A066211 * A066213 A066214 A066215
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KEYWORD
| sign
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AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Dec 17 2001
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