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A356664
Numbers k such that A225205(k) is in A354549.
2
0, 2, 4, 10, 12, 14, 18, 20, 22, 30, 32, 34, 38, 40, 44, 48, 52, 60, 62, 72, 76, 78, 80, 82, 92, 94, 100, 104, 116, 120, 126, 130, 132, 134, 138, 140, 142, 144, 146, 148, 152, 154, 156, 158, 160, 168, 176, 180, 182, 186, 188, 192, 194, 202, 210, 222, 224, 226, 228, 230, 232
OFFSET
1,2
COMMENTS
Numbers k such that floor(A225205(k)^2*phi) = A225204(k)^2, phi = A001622.
Even numbers k such that (A225204(k)^2+1)/A225205(k)^2 > phi.
Even k is a term in A356591 if and only k is in this sequence and {A225205(k)^2*phi} < phi^(-2), where {} denotes the fractional part; see the comments in A354513.
Conjecture: this and A356591 have the same natural density.
LINKS
FORMULA
A354549(n+1) = A225205(a(n)).
EXAMPLE
4 is a term because A225204(4) = 125 and A225205(4) = 159, and floor(125^2*phi) = 159^2.
PROG
(PARI) print1("0, "); 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(n%2 == 1 && (conv[1, n]^2+1)^2 - (conv[1, n]^2+1)*(conv[2, n]^2) - (conv[2, n]^2)^2 > 0, print1(n-1, ", ")))
\\ Here conv[1, n] = A225204(n-1), conv[2, n] = A225205(n-1), n odd implies conv[1, n]/conv[2, n] < sqrt((1+sqrt(5))/2); let A = conv[1, n]^2+1, B = conv[2, n]^2, then A^2 - A*B - B^2 > 0 implies A/B > (1+sqrt(5))/2
\\ 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