%I #45 Aug 28 2022 08:28:51
%S 3,5,7,15,19,20,25,27,34,37,40,44,47,48,52,57,65,77,89,91,92,100,105,
%T 107,111,121,123,126,127,129,138,141,153,163,165,167,171,173,179,182,
%U 183,185,189,193,195,202,205,209,211,213,215,222,224,226,230,232,234,236,238
%N Numbers k such that A225205(k) is in A354513.
%C Numbers k such that floor((A225205(k)^2+1)*phi) = A225204(k)^2+1, phi = A001622.
%C Numbers k such that (A225204(k)^2+1)/(A225205(k)^2+1) < phi < (A225204(k)^2+2)/(A225205(k)^2+1).
%C Conjecture: the odd numbers (numbers k such that A225204(k)/A225205(k) > sqrt(phi)) have relative density phi^(-1), and the even numbers (number k such that A225204(k)/A225205(k) < sqrt(phi)) have relative density phi^(-2). It is conjectured so because we have lim_{k->+oo} (m/k - sqrt((m^2+1)/(k^2+1)))/(sqrt((m^2+2)/(k^2+1)) - m/k) = phi if m/k -> sqrt(phi).
%C Even k is a term if and only floor(A225205(k)^2*phi) = A225204(k)^2 (k is in A356664) and {A225205(k)^2*phi} < phi^(-2), where {} denotes the fractional part; see the comments in A354513.
%H Jianing Song, <a href="/A356591/b356591.txt">Table of n, a(n) for n = 1..283</a>
%F A354513(n) = A225205(a(n)).
%e 3 is a term because A225204(3) = 14 and A225205(3) = 11, and floor((11^2+1)*phi) = 14^2+1.
%o (PARI) A000201(n) = (n+sqrtint(5*n^2))\2;
%o my(cofr=A331692_vector_bits(1000), conv=matrix(2, #cofr)); conv[, 1]=[1, 1]~; conv[, 2]=[4, 3]~; for(n=3, #cofr, conv[, n]=cofr[n]*conv[, n-1]+conv[, n-2]; if(A000201(conv[2, n]^2+1) == conv[1, n]^2+1, print1(n-1, ", ")))
%o \\ Here conv[1, n] = A225204(n-1), conv[2, n] = A225205(n-1)
%o \\ Modified by _Jianing Song_, Aug 28 2022 according to _Kevin Ryde_'s program for A331692
%Y Cf. A001622, A225204, A225205, A354513, A356664.
%K nonn
%O 1,1
%A _Jianing Song_, Aug 21 2022