cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A356591 Numbers k such that A225205(k) is in A354513.

Original entry on oeis.org

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

Views

Author

Jianing Song, Aug 21 2022

Keywords

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.

Examples

			3 is a term because A225204(3) = 14 and A225205(3) = 11, and floor((11^2+1)*phi) = 14^2+1.
		

Crossrefs

Programs

  • 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

Formula

A354513(n) = A225205(a(n)).

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Aug 21 2022

Keywords

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.

Examples

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

Crossrefs

Programs

  • 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

Formula

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

A331692 Continued fraction for square root of golden ratio.

Original entry on oeis.org

1, 3, 1, 2, 11, 3, 5, 1, 10, 1, 2, 5, 3, 5, 1, 2, 4, 1, 1, 2, 4, 51, 1, 4, 2, 2, 31, 1, 3, 1, 1, 5, 1, 1, 1, 14, 1, 1, 4, 2, 2, 8, 1, 1, 3, 23, 1, 1, 4, 16, 2, 1, 2, 13, 2, 1, 1, 1, 3, 1, 1, 4, 3, 2, 1, 1, 36, 1, 2, 1, 1, 1, 2, 3, 2, 1, 3, 3, 31, 2, 1, 2, 2, 2
Offset: 0

Views

Author

Jordan Paschke, Mar 03 2020

Keywords

Examples

			1 + 1/(3 + 1/(1 + 1/(2 + 1/(11 + ...)))) = sqrt(phi).
		

Crossrefs

Cf. A001622, A139339 (decimal expansion).
Cf. A225204/A225205 (convergents).

Programs

  • Mathematica
    ContinuedFraction[Sqrt[(1 + Sqrt[5])/2], 1000]
  • PARI
    contfrac(sqrt((1+sqrt(5))/2)) \\ Michel Marcus, Mar 04 2020
    
  • PARI
    \\ See Ryde link

A225204 Numerators of convergents to the square root of the golden ratio.

Original entry on oeis.org

1, 4, 5, 14, 159, 491, 2614, 3105, 33664, 36769, 107202, 572779, 1825539, 9700474, 11526013, 32752500, 142536013, 175288513, 317824526, 810937565, 3561574786, 182451251651, 186012826437, 926502557399, 2039017941235, 5004538439869, 157179709577174
Offset: 0

Views

Author

Eric W. Weisstein, May 01 2013

Keywords

Examples

			1, 4/3, 5/4, 14/11, 159/125, 491/386, 2614/2055, 3105/2441, ... = A225204/A225205
		

Crossrefs

Cf. A001622, A139339, A331692, A225205 (denominators).

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[GoldenRatio], 20]]

Formula

a(n) = A331692(n)*a(n-1) + a(n-2) for n >= 2. - Jianing Song, Aug 18 2022

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

Views

Author

Chittaranjan Pardeshi, Aug 16 2022

Keywords

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, ...
		

Crossrefs

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

A354549 Numbers k such that floor(k^2*phi) is a square, where phi = A001622 is the golden ratio.

Original entry on oeis.org

0, 1, 4, 125, 84277, 1435150, 9061191, 249858189, 2799936925, 146234239784, 1139643680683264, 7471434609455791, 21274660147684109, 2911209509190673141, 15845190736671957299, 995980378496501932493, 213688560255016550712685, 28372206851301867342910959
Offset: 1

Views

Author

Jianing Song, Aug 18 2022

Keywords

Comments

Numbers k such that there exists m such that m^2 <= k^2*phi < m^2+1. For k > 0, m/k is a lower convergent to sqrt(phi) since m/k < sqrt(phi) < sqrt(m^2+1)/k, so |sqrt(phi) - m/k| < (sqrt(m^2+1)-m)/k < 1/(2*k^2) (see the Mathematics Stack Exchange link). As a result, this is a subsequence of {A225205(2*n): n>=0}. The terms > 0 are b(0), b(2), b(4), b(10), b(12), b(14), ... for b = A225205. Note that sqrt(phi) - A225204(2*r)/A225205(2*r) < 1/(A225205(2*r)*A225205(2*r+1)) (by Theorem 5 of the Wikipedia link), so A225205(2*r) is a term if sqrt(A225204(2*r)^2+1) + A225204(2*r) < A225205(2*r+1).
Consider the numbers s such that A035513(s,0) = A000201(s) = floor(s*phi) and A035513(s,1) = A003622(s) = floor(floor(s*phi)*phi) are both squares. If s has this property, then clearly floor(s*phi) is the square of a term. However, for k > 0 being a term, k^2 is not always in A000201. This happens if and only if {k^2*phi} < phi^(-2), where {} denotes the fractional part (see A000201); for example k = 2799936925, for which floor(k^2*phi) = 3561574786^2. In this case, floor(k^2*phi) is not in A003622 since n -> floor(n*phi) is an injection. Suppose that A035513(p,q) = floor(k^2*phi) for p >= 1, q >= 2, then A035513(p,q-1) = floor(k^2*phi) since floor((k^2+1)*phi) = floor(k^2*phi)+1 in this case.

Examples

			4 is a term since floor(4^2*phi) = 5^2.
125 is a term since floor(125^2*phi) = 159^2.
		

Crossrefs

Programs

  • PARI
    A000201(n) = (n+sqrtint(5*n^2))\2; isok(n) = issquare(A000201(n^2))
    
  • PARI
    print1("0, 1, "); 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(conv[2,n], ", ")))
    \\ 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
Showing 1-6 of 6 results.