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A356591
Numbers k such that A225205(k) is in A354513.
2
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
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
KEYWORD
nonn
AUTHOR
Jianing Song, Aug 21 2022
STATUS
approved