A066212 A measure of how close the golden ratio is to rational numbers.

-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

Offset: 1

Author

Henry Bottomley, Dec 17 2001

Keywords

• sign

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.

Examples

			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.
		

Links

• Harry J. Smith, Table of n, a(n) for n = 1..1000

Programs

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

Formula

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