A371625 The x-coordinate of the point (x,y) where x + y = n, x is an integer, and x/y is as close as possible to phi (by absolute difference).

0, 1, 2, 2, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 10, 10, 11, 12, 12, 13, 14, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 23, 24, 25, 25, 26, 27, 27, 28, 28, 29, 30, 30, 31, 32, 32, 33, 33, 34, 35, 35, 36, 36, 37, 38, 38, 39, 40, 40, 41, 41, 42, 43

Offset: 1

Author

Colin Linzer, Mar 29 2024

Keywords

• nonn

Comments

a(n) = x = either ceiling or floor of n/phi, according to which minimizes abs(x/(n-x) - phi).

Each term is equal to or one greater than the previous term.

The average run length approaches phi.

The 2 following statements are equivalent for any real n and any function f(x) such that for any real x, f(x) equals an integer within the range (x-1,x+1) (e.g., round(x), ceiling(x), floor(x)):

A371626(n) != A371627(n);

A371626(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 floor(n/phi) to being equal to 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(2w+1)+sqrt((2w+1)^2 * phi^-4 + 4/phi)) solved for w. The equation gives the n-value of w-th swap from a(n) = floor(n/phi) to a(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 times that a(n) swapped from being equal to ceiling(n/phi) to being equal to floor(n/phi) when n is extended to the reals.

Examples

			For n=5, the possibilities are (0,5), (1,4), (2,3), (3,2) and (4,1). Of those, 3/2 is the closest to phi, so a(5)=3.
		

Crossrefs

Cf. A001622 (phi), A371626 (y_coordinate), A371627 (with 1/phi), A002163 (sqrt(5)).

Formula

a(n) = n - A371626(n).

a(n) = ceiling(n/phi) if floor(s(n)) + floor(n/phi) is even.

a(n) = floor(n/phi) if floor(s(n)) + floor(n/phi) is odd.

a(n) = ceiling(n/phi) - (floor(s(n))+floor(n/phi) mod 2).

a(n) = round(n/phi) + floor(s(n)) - floor(n/phi+1/2).