cp's OEIS Frontend

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.

%I A371627 #15 Aug 05 2025 07:50:26
%S A371627 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,
%T A371627 12,13,13,13,14,14,15,15,15,16,16,16,17,17,18,18,18,19,19,19,20,20,21,
%U A371627 21,21,22,22,23,23,23,24,24,24,25,25,26,26,26,27,27,27
%N 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.
%C A371627 Each term is equal to or one greater than the previous term.
%C A371627 The average run length approaches 1+phi.
%C A371627 a(n) = x = either ceiling or floor of n/phi^2, according to which minimizes abs(x/(n-x) - phi).
%C A371627 The 4 following statements are equivalent for any positive integer n and any function f(x) such that for all real x, x-1<f(x)<x+1 and f(x) is integral, where (y(n)) is the sequence for the y coordinate:
%C A371627   a(n) != A371626(n);
%C A371627   A371625(n) != y(n);
%C A371627   a(n) != n-f(n/phi) xor A371626(n) != n-f(n/phi);
%C A371627   A371625(n) != f(n/phi) xor y(n) != f(n/phi).
%e A371627 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.
%Y A371627 Cf. A094214 (1/phi), A371625 (with phi).
%K A371627 nonn,frac
%O A371627 1,5
%A A371627 _Colin Linzer_, Mar 29 2024
%E A371627 Elements referring to sequences that were not submitted removed by _Peter Munn_, Aug 04 2025