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.

Previous Showing 11-20 of 20 results.

A244012 Numerators of rational approximations to sqrt(7) obtained from Newton's method.

Original entry on oeis.org

2, 11, 233, 108497, 23543191457, 1108563727961872518977, 2457827077905448997994482872789298261401217, 12081827889770476116093110581355561229584727594431650162181251776430351279198649072897
Offset: 0

Views

Author

N. J. A. Sloane, Jun 18 2014

Keywords

Examples

			2, 11/4, 233/88, 108497/41008, 23543191457/8898489952, ...

Links

Crossrefs

Cf. A244013 (denominators).
The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

Programs

  • Maple
    N:=7;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A244013 Denominators of rational approximations to sqrt(7) obtained from Newton's method.

Original entry on oeis.org

1, 4, 88, 41008, 8898489952, 418997705236253480128, 928971316248341903257187589777603944778112, 4566501711345281867283814391125123371716411674583075407993026856131137508750543524608
Offset: 0

Views

Author

N. J. A. Sloane, Jun 18 2014

Keywords

Examples

			2, 11/4, 233/88, 108497/41008, 23543191457/8898489952, ...

Links

Crossrefs

Cf. A244012 (numerators).
The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

Programs

  • Maple
    N:=7;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A244014 Numerators of rational approximations to sqrt(6) obtained from Newton's method.

Original entry on oeis.org

2, 5, 49, 4801, 46099201, 4250272665676801, 36129635465198759610694779187201, 2610701117696295981568349760414651575095962187244375364404428801
Offset: 0

Views

Author

N. J. A. Sloane, Jun 18 2014

Keywords

Examples

			2, 5/2, 49/20, 4801/1960, 46099201/18819920, ...

Crossrefs

Cf. A244015, A084765.
The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

Programs

  • Maple
    N:=6;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A244015 Denominators of rational approximations to sqrt(6) obtained from Newton's method.

Original entry on oeis.org

1, 2, 20, 1960, 18819920, 1735166549767840, 14749861913749949808286047759680, 1065814268211609269094400465471990022332221793124358274759711360
Offset: 0

Views

Author

N. J. A. Sloane, Jun 18 2014

Keywords

Examples

			2, 5/2, 49/20, 4801/1960, 46099201/18819920, ...

Crossrefs

Cf. A244014 (numerators).
The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

Programs

  • Magma
    m:=9; f:=[n eq 1 select 2 else (Self(n-1)+6/Self(n-1))/2: n in [1..m]]; [Denominator(f[n]): n in [1..m]]; // Vincenzo Librandi, Jan 12 2016
  • Maple
    N:=6;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A177879 The smallest odd prime factor of the Lucas-Lehmer number A003010(n).

Original entry on oeis.org

7, 97, 31, 708158977, 127, 22783, 113210499946729046527, 12289, 1049179854847, 22427452848394140276947044397991663611794141183, 8191, 29687809
Offset: 1

Views

Author

G. L. Honaker, Jr., Dec 13 2010

Keywords

Comments

Does this sequence include all of the Mersenne primes greater than 3?
a(8)=12289; a(11)=8191, a(15)=131071.
Also the least prime factor of A002812(n). - Michel Marcus, Dec 16 2022
a(p-2) = 2^p-1 for all odd Mersenne exponents p in A000043? - Thomas Ordowski, Aug 12 2018

Examples

			A003010(3)=37634 and its smallest odd prime factor is 31.

Links

Crossrefs

Cf. A000040, A003010, A000668, A002812.

Extensions

More terms, using factordb, from Michel Marcus and Hugo Pfoertner, Dec 16 2022

A094680 a(n+1) = 4*a(n)^3 - 3*a(n), with a(0) = 2.

Original entry on oeis.org

2, 26, 70226, 1385331749802026, 10634604778476758291777057017318241822792488226
Offset: 0

Views

Author

Jose Eduardo Blazek, Jun 07 2004

Keywords

Comments

Smallest positive integer x satisfying the Pell equation x^2 - 3^(2*n-3) * y^2 = 1. - A.H.M. Smeets, Sep 29 2017
Term a(5) has 139 decimal digits and a(6) has 417 decimal digits. - Andrew Howroyd, Feb 25 2018

Links

Crossrefs

Cf. A001075, A002812.

Programs

  • Mathematica
    NestList[4 #^3 - 3 # &, 2, 5] (* Michael De Vlieger, Oct 02 2017 *)
  • PARI
    a(n) = if (n==0, 2, 4*a(n-1)^3 - 3*a(n-1)); \\ Michel Marcus, Oct 03 2017
  • PARI
    a(n) = polchebyshev(3^n, 1, 2); \\ Michel Marcus, Oct 03 2017

Formula

a(n) = cosh(3^n*arccosh(2)).
a(n) = ChebyshevT(3^n, 2). - Vladeta Jovovic, Jun 11 2004
From A.H.M. Smeets, Oct 02 2017: (Start)
a(n) = A001075(3^(n-2))
a(n) = A002350(3^(2n-3)). (End)

Extensions

More terms from Vladeta Jovovic, Jun 11 2004
Offset corrected by Michel Marcus, Oct 03 2017

A382437 a(n) = a(n-1)^2 + 4 * a(n-1), with a(0) = 2.

Original entry on oeis.org

2, 12, 192, 37632, 1416317952, 2005956546822746112, 4023861667741036022825635656102100992, 16191462721115671781777559070120513664958590125499158514329308740975788032
Offset: 0

Views

Author

V. Barbera, Mar 25 2025

Keywords

Links

Crossrefs

Cf. A002812, A003010.

Programs

  • Mathematica
    NestList[#*(4 + #) &, 2, 8] (* Paolo Xausa, Apr 01 2025 *)
  • PARI
    a(n)=if(n, a(n-1)^2 + 4*a(n-1), 2);
vector(8, i, a(i-1))

Formula

a(n) = A003010(n) - 2.
a(n)/2 = A002812(n) - 1.
For n > 1: a(n) = 3 * 2^(2*n) * Product_{i = 0..n-2} A002812(i)^2.
Conjecture: a(n) = Sum_{k=1..2^n} (2^n * 2^k * binomial(2^n + k - 1, 2*k - 1) / k).

A213680 a(n) = 2*a(n-1)^2/3-3 with a(0)=6.

Original entry on oeis.org

6, 21, 291, 56451, 2124476931, 3008934820234119171, 6035792501611554034238453484153151491, 24287194081673507672666338605180770497437885188248737771493963111463682051
Offset: 0

Views

Author

Jose Eduardo Blazek, Mar 04 2013

Keywords

Examples

			a(1) = 2*a(0)^2/3-3 = 2*6^2/3-3 = 21,
a(2) = 2*a(1)^2/3-3 = 2*21^2/3-3 = 291,
a(3) = 2*a(2)^2/3-3 = 2*291^2/3-3 = 2*84681/3-3 = 56451.
Or, by the first formula:
a(3) = 3*cosh(2^3*arccosh(2)) = 56451,
a(4) = 3*cosh(2^4*arccosh(2)) = 2124476931.

Crossrefs

Cf. A002812.

Programs

  • Sage
    var('x')
p=2*x^2/3-3
s=[6]
for i in [0..8]:
    s=s+[p(s[i])]
show(s)

Formula

a(n) = 3*cosh(2^n*arccosh(2)).
a(n) = 3*A002812(n). [Giovanni Resta, Mar 04 2013]

A242232 a(n) = 2*a(n-1)^2 - 1, a(0)=6.

Original entry on oeis.org

6, 71, 10081, 203253121, 82623662392481281, 13653339174293451118767199870801921, 372827341216592355174245573447441869623455324379507680549087234580481
Offset: 0

Views

Author

Vaclav Kotesovec, May 08 2014

Keywords

Comments

In general, for a(0)=p is a(n) = cosh(2^n*arccosh(p)) = (1/2)*(p+sqrt(p^2-1))^(2^n) + (1/2)*(p+sqrt(p^2-1))^(-2^n).

Crossrefs

Cf. A002812, A001601, A005828, A084765.

Programs

  • Mathematica
    RecurrenceTable[{a[n+1]==2*a[n]^2-1,a[0]==6},a,{n,0,10}]
NestList[2#^2-1&,6,10] (* Harvey P. Dale, Jun 12 2025 *)

Formula

a(n) = (1/2)*(6+sqrt(35))^(2^n) + (1/2)*(6+sqrt(35))^(-2^n).
a(n) = A023038(2^n).
a(n) = T(2^n,6), where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. - Peter Bala, Mar 30 2022

A261532 Hall's sequence g_n.

Original entry on oeis.org

7, 589, 11064985, 7835767761026353, 7859104819806710982081319640824417, 15811975313589523224392147529414564125936123169432771986649715567359169
Offset: 3

Views

Author

Joerg Arndt, Aug 24 2015

Keywords

Links

Crossrefs

Cf. A002812, A003010.

Programs

  • Magma
    [n le 1 select 7 else 2*(Self(n-1)-1)*(1+2^(n+1)*(Self(n-1)-1)) + 1: n in [1..6]]; // Vincenzo Librandi, Aug 24 2015
  • PARI
    N=11;  g=vector(N);  g[3] = 7;
for (n=3, N-1, g[n+1] = 2*(g[n]-1)*(1+2^n*(g[n]-1))+1 );
vector(N-3, j, g[j+2])

Formula

a(n) = g(n) where g(3) = 7 and g(n+1) = 2*(g(n)-1)*(1+2^n*(g(n)-1)) + 1.
Previous Showing 11-20 of 20 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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