A371625 -id:A371625 - cp's OEIS frontend

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).

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Offset: 1

Colin Linzer, Mar 29 2024

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 2 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);
  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.

			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.

Cf. A001622 (phi), A371625 (x_coordinate).

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).

Elements referring to sequences that were not submitted removed by Peter Munn, Aug 04 2025

A371627 The x-coordinate of the point where x + y = n, x is an integer and x/y is as close as possible to 1/phi.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 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, 27

Offset: 1

Colin Linzer, Mar 29 2024

Each term is equal to or one greater than the previous term.
The average run length approaches 1+phi.
a(n) = x = either ceiling or floor of n/phi^2, according to which minimizes abs(x/(n-x) - phi).
The 4 following statements are equivalent for any positive integer n and any function f(x) such that for all real x, x-1  a(n) != A371626(n);
  A371625(n) != y(n);
  a(n) != n-f(n/phi) xor A371626(n) != n-f(n/phi);
  A371625(n) != f(n/phi) xor y(n) != f(n/phi).

			For n=4, the possibilities are (0,4), (1,3), (2,2), and (3,1). 1/3 is the closest to 1/phi out of them, so a(4)=1.

Cf. A094214 (1/phi), A371625 (with phi).

Elements referring to sequences that were not submitted removed by Peter Munn, Aug 04 2025

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cp's OEIS Frontend

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).

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