A356664 -id:A356664 - cp's OEIS frontend

3, 5, 7, 15, 19, 20, 25, 27, 34, 37, 40, 44, 47, 48, 52, 57, 65, 77, 89, 91, 92, 100, 105, 107, 111, 121, 123, 126, 127, 129, 138, 141, 153, 163, 165, 167, 171, 173, 179, 182, 183, 185, 189, 193, 195, 202, 205, 209, 211, 213, 215, 222, 224, 226, 230, 232, 234, 236, 238

Offset: 1

Jianing Song, Aug 21 2022

nonn

Numbers k such that floor((A225205(k)^2+1)*phi) = A225204(k)^2+1, phi = A001622.
Numbers k such that (A225204(k)^2+1)/(A225205(k)^2+1) < phi < (A225204(k)^2+2)/(A225205(k)^2+1).
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).
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.

			3 is a term because A225204(3) = 14 and A225205(3) = 11, and floor((11^2+1)*phi) = 14^2+1.

Jianing Song, Table of n, a(n) for n = 1..283

Cf. A001622, A225204, A225205, A354513, A356664.

A000201(n) = (n+sqrtint(5*n^2))\2;
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, ", ")))
\\ Here conv[1, n] = A225204(n-1), conv[2, n] = A225205(n-1)
\\ Modified by Jianing Song, Aug 28 2022 according to Kevin Ryde's program for A331692

A354513(n) = A225205(a(n)).

cp's OEIS Frontend

A356591 Numbers k such that A225205(k) is in A354513.

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