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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).
2
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
1,3
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 4 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.
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)
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), A371626 (y_coordinate), A371627 (with 1/phi), A002163 (sqrt(5)).
Sequence in context: A076935 A019446 A097369 * A257808 A249036 A096607
KEYWORD
nonn
AUTHOR
Colin Linzer, Mar 29 2024
STATUS
approved