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

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

Offset: 1

Author

Chittaranjan Pardeshi, Aug 16 2022

Keywords

• nonn

Comments

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)

Examples

			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, ...
		

Links

• 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

Crossrefs

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

Programs

PARI

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

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