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

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

1,1

Comments

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.

Links

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

Formula

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

Example

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

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A160690 A085494 A348441 * A029460 A277910 A029466

Adjacent sequences: A356588 A356589 A356590 * A356592 A356593 A356594

Keyword

nonn

Author

Jianing Song, Aug 21 2022

Status

approved