A354549 -id:A354549 - cp's OEIS frontend

A356664 Numbers k such that A225205(k) is in A354549.

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

Jianing Song, Aug 21 2022

nonn

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.

			4 is a term because A225204(4) = 125 and A225205(4) = 159, and floor(125^2*phi) = 159^2.

Jianing Song, Table of n, a(n) for n = 1..293

Cf. A001622, A225204, A225205, A354549, A354513, A356591.

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

A354549(n+1) = A225205(a(n)).

A354513 The numbers whose square's position in the Wythoff array is immediately followed by another square in the next column.

Original entry on oeis.org

11, 386, 2441, 25748423, 637519684, 2799936925, 3934324789543, 127501370029150, 21274660147684109, 644571595359295797, 15845190736671957299, 995980378496501932493, 47375682236837399943653, 213688560255016550712685, 28372206851301867342910959, 3120729065082950391169492805

Offset: 1

Chittaranjan Pardeshi, Aug 16 2022

nonn

From Jianing Song, Aug 21 2022: (Start)
Numbers k > 0 such that floor((k^2+1)*phi) - 1 is a square, phi = A001622.
Suppose that k is a term and that floor((k^2+1)*phi) = m^2+1, then (m^2+1)/(k^2+1) < phi < (m^2+2)/(k^2+1), so |sqrt(phi) - m/k| < max{m/k - sqrt((m^2+1)/(k^2+1)), sqrt((m^2+2)/(k^2+1)) - m/k} = m/k - sqrt((m^2+1)/(k^2+1)) <= sqrt((k^2+1)*phi-1)/k - sqrt(phi) < 1/(2*sqrt(phi)*k^2). According to the Mathematics Stack Exchange link, m/k is a convergent to sqrt(phi), so this is a subsequence of A225205. The terms are b(3), b(5), b(11), b(15), b(19), b(20), ... for b = A225205.
For k = A225205(r), m = A225204(r), we have |sqrt(phi) - m/k| < 1/(k*A225205(r+1)) (by Theorem 5 of the Wikipedia link), so k = A225205(r) is a term if 1/(k*A225205(r+1)) < min{m/k - sqrt((m^2+1)/(k^2+1)), sqrt((m^2+2)/(k^2+1)) - m/k} = sqrt((m^2+2)/(k^2+1)) - m/k, or A225205(r+1) > (k*sqrt((m^2+2)/(k^2+1)) - m)^(-1).
If k = A225205(r) is a term with even r, then k is also in A354549, since m^2 < k^2*phi < k^2*(m^2+2)/(k^2+1) < m^2+phi^(-2) for m = A225204(r), so floor(k^2*phi) = m^2. Furthermore we have {k^2*phi} < phi^(-2), where {} denotes the fractional part. Conversely, if k is in A354549 and {k^2*phi} < phi^(-2), then k is in this sequence since floor((k^2+1)*phi) = floor(k^2*phi)+1 in this case. (End)

			11 is a term since 11^2 = 121 has another square, 196 = 14^2, immediately to its right in the Wythoff array. Array row: 46, 75, 121, 196, ...

Jianing Song, Table of n, a(n) for n = 1..127
Mathematics Stack Exchange, If |x - p/q| < 1/(2*q^2) then p/q is necessarily one of the convergents
Wikipedia, Continued fraction

Cf. A035513, A026274, A352538, A001622, A225204, A225205, A354549.

phi=quadgen(5);
nextcolumn(x) = ((x+1)*phi-1)\1; \\ A026274(x+1)
for(i=1, 10000000000, if ( issquare( nextcolumn (i^2)), print1(i, ", ")));

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(conv[2, n], ", "))) \\ Jianing Song, Aug 21 2022, modified on Aug 28 2022 according to Kevin Ryde's program for A331692

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A356664 Numbers k such that A225205(k) is in A354549.

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A354513 The numbers whose square's position in the Wythoff array is immediately followed by another square in the next column.

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