# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a354513 Showing 1-1 of 1 %I A354513 #64 Oct 06 2024 12:25:33 %S A354513 11,386,2441,25748423,637519684,2799936925,3934324789543, %T A354513 127501370029150,21274660147684109,644571595359295797, %U A354513 15845190736671957299,995980378496501932493,47375682236837399943653,213688560255016550712685,28372206851301867342910959,3120729065082950391169492805 %N A354513 The numbers whose square's position in the Wythoff array is immediately followed by another square in the next column. %C A354513 From _Jianing Song_, Aug 21 2022: (Start) %C A354513 Numbers k > 0 such that floor((k^2+1)*phi) - 1 is a square, phi = A001622. %C A354513 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. %C A354513 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). %C A354513 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) %H A354513 Jianing Song, Table of n, a(n) for n = 1..127 %H A354513 Mathematics Stack Exchange, If |x - p/q| < 1/(2*q^2) then p/q is necessarily one of the convergents %H A354513 Wikipedia, Continued fraction %e A354513 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, ... %o A354513 (PARI) %o A354513 phi=quadgen(5); %o A354513 nextcolumn(x) = ((x+1)*phi-1)\1; \\ A026274(x+1) %o A354513 for(i=1, 10000000000, if ( issquare( nextcolumn (i^2)), print1(i, ", "))); %o A354513 (PARI) A000201(n) = (n+sqrtint(5*n^2))\2; %o A354513 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 %Y A354513 Cf. A035513, A026274, A352538, A001622, A225204, A225205, A354549. %K A354513 nonn %O A354513 1,1 %A A354513 _Chittaranjan Pardeshi_, Aug 16 2022 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE