|
|
A371626
|
|
The y-coordinate of the point where x + y = n, x is an integer and x/y is as close as possible to phi (by absolute difference).
|
|
2
|
|
|
1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 10, 10, 10, 11, 11, 11, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 28
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,4
|
|
COMMENTS
|
a(n) = k = n - either ceiling or floor of n/phi, according to which minimizes abs((n-k)/k - phi).
Each term is equal to or one greater than the previous term.
The average run length approaches 1+phi.
The 4 following statements are equivalent for any positive integer n and any function f(x) such that for any real x, f(x) equals a integer within the range (x-1,x+1):
a(n) != n-f(n/phi) xor A371627(n) != n-f(n/phi);
Let s(n) = (phi*n - 1 - sqrt(1+(n^2)*(phi^-4)))/2.
Floor(s(n)) equals the number of times that a(n) swapped from being equal to n-floor(n/phi) to being equal to n-ceiling(n/phi) when n is extended to the reals.
This is true because s(n) is the solution to the equation n = (phi/4) * (phi(2r+1) + sqrt((2r+1)^2 / phi^4 + 4/phi)) solved for w. The equation gives the n-value of w-th swap from a(n) = n-floor(n/phi) to a(n) = n-ceiling(n/phi).
s(n) is asymptotic to n/phi - 1/2.
Floor(s(n)) != floor(n/phi - 1/2) <-> a(n) != round(n).
Floor(n/phi) equals the number of that a(n) swapped from being equal to n-ceiling(n/phi) to being equal to n-floor(n/phi) when n is extended to the reals.
|
|
LINKS
|
|
|
FORMULA
|
Let s(n) = (phi*n - 1 - sqrt(1+(n^2) / phi^4))/2.
Floor(s(n))+floor(n/phi) is even -> a(n) = n-ceiling(n/phi) = (n mod 1) + floor(n/phi^2).
Floor(s(n))+floor(n/phi) is odd -> a(n) = n-floor(n/phi) = (n mod 1) + ceiling(n/phi^2).
a(n) = -a(-n).
|
|
EXAMPLE
|
For n=5, the possibilities are (0,5), (1,4), (2,3), (3,2), & (4,1). Of those, 3/2 is the closest to phi, so a(5)=3.
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|