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A measure of how close the golden ratio is to rational numbers.
3

%I #12 Aug 18 2024 19:03:12

%S -3,8,-21,8,55,-21,21,-144,-21,55,-55,29,377,-40,55,-144,-34,144,-74,

%T 55,-987,-55,107,-144,55,377,-86,92,-377,-65,195,-144,84,2584,-95,144,

%U -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

%N A measure of how close the golden ratio is to rational numbers.

%C 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.

%H Harry J. Smith, <a href="/A066212/b066212.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = round(1/(phi-round(phi*n)/n)) where phi = (sqrt(5)+1)/2 = 1.6180339... = round(1/(phi-A007067(n)/n)).

%e 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.

%o (PARI) { phi=(1 + sqrt(5))/2; for (n=1, 1000, a=round(1/(phi - round(phi*n)/n)); write("b066212.txt", n, " ", a) ) } \\ _Harry J. Smith_, Feb 06 2010

%K sign

%O 1,1

%A _Henry Bottomley_, Dec 17 2001