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).

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

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) != A371627(n);

A371625(n) != A371628(n);

a(n) != n-f(n/phi) xor A371627(n) != n-f(n/phi);

A371625(n) != f(n/phi) xor A371628(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

Table of n, a(n) for n=1..72.

Formula

a(n) = n - A371625(n).

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

Cf. A001622 (phi), A371625 (x_coordinate), A371628 (with 1/phi), A371630 (with -1/phi).

Sequence in context: A057367 A032634 A057366 * A189663 A341440 A355028

Adjacent sequences: A371623 A371624 A371625 * A371627 A371628 A371629

Keyword

nonn,frac

Author

Colin Linzer, Mar 29 2024

Status

approved