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 1-3 of 3 results.

A262026 The positive odd fundamental solutions y = y0(n) for the Pell equation x^2 - d*y^2 = +1. It turns out that d = d(n) coincides with A007970(n).

Original entry on oeis.org

1, 3, 1, 3, 1, 39, 5, 1, 5, 273, 3, 1, 3, 531, 7, 1, 7, 69, 1, 5967, 413, 3, 9, 1, 9, 3, 21, 165, 5, 1, 22419, 5, 93, 105, 11, 1, 11, 419775, 51, 927, 21, 3, 6578829, 1, 140634693, 3, 105, 57, 5019135, 13, 1, 13, 153, 15, 313191, 123, 650783, 7, 1, 1153080099, 7, 45, 19162705353, 3, 33, 5
Offset: 1

Views

Author

Wolfdieter Lang, Oct 04 2015

Keywords

Comments

The corresponding x = x0(n) values are given by A262027(n).
This is a proper subset of A033317 corresponding to its odd members.
For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970.
For the positive even fundamental solutions y = y0(n) of x^2 - d*y^2 = 1, where d = d(n) coincides with A007969(n) see 2*A261250(n).
If d(n) = A007970(n) is odd (necessarily congruent to 3 modulus 4) then x0(n) is even, and if d(n) is even (necessarily congruent to 0 modulus 8) then x0 is odd.

Examples

			The first triples [d(n), x0(n), y0(n)] are: [3,2,1], [7,8,3], [8,3,1], [11,10,3], [15,4,1], [19,170,39], [23,24,5], [24,5,1], [27,26,5], [31,1520,273], [32,17,3], [35,6,1], [40,19,3], [43,3482,531], [47,48,7], [48,7,1], [51,50,7], [59,530,69], [63,8,1], [67,48842,5967], [71,3480,413], [75,26,3], [79,80,9], [80,9,1], [83,82,9], [87,28,3], [88,197,21], [91,1574,165], [96,49,5], [99,10,1], [103,227528,22419], ...

Crossrefs

Cf. A007970, A033317, A262027, A262028.

Formula

x0(n)^2 - d(n)*a(n)^2 = +1 with x0(n) =
A262027(n) and d(n) = A007970(n). (x0(n), y0(n) = a(n)) are the positive fundamental solutions of this Pell equation x^2 - d*y^2 = +1 with odd y = y0.

A263008 First member T0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).

Original entry on oeis.org

1, 1, 1, 3, 1, 13, 1, 1, 5, 7, 1, 1, 3, 59, 1, 1, 7, 23, 1, 221, 7, 1, 1, 1, 9, 3, 7, 11, 1, 1, 47, 5, 31, 15, 1, 1, 11, 193, 3, 103, 3, 1, 8807, 1, 3383, 3, 21, 3, 8005, 1, 1, 13, 17, 3, 2047
Offset: 1

Views

Author

Wolfdieter Lang, Oct 29 2015

Keywords

Comments

The 2-happy numbers D(n)*E(n) are given in A007970(n) (called rhombic numbers in the Conway paper). D(n) = A191856(n), E(n) = A191857(n). Here the corresponding smallest positive numbers satisfying E(n)*U(n)^2 - D(n)*T(n)^2 = +2, n >= 1, with odd U(n) and T(n) are given as T0(n) = a(n) and U0(n) = A263009(n).
In the W. Lang link the first U0(n) and T0(n) numbers are given in the Table for d(n) = A007970(n), n >= 1.
In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = T0(n) numbers appear for the t = 2 rows in column v.

Examples

			n = 6: 2-happy number A007970(6) = 19 = 1*19 = A191856(6)*A191857(6). 19*A263009(6)^2 - 1*a(6)^2 = 19*3^2 - 1*13^2 = +2. This is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).

Links

Crossrefs

Cf. A007970, A191856, A191857, A191860, A263009, A262026, A262027, A262028.

Formula

A191857(n)*A263009(n)^2 - A191856(n)*a(n)^2 = +2, and a(n) with A263009(n) is the smallest positive solution for the given 2-happy couple (A191856(n), A191857(n)).

A263009 Second member U0(n) of the smallest positive pair (T0(n), U0(n)) for the n-th 2-happy number couple (D(n), E(n)).

Original entry on oeis.org

1, 3, 1, 1, 1, 3, 5, 1, 1, 39, 3, 1, 1, 9, 7, 1, 1, 3, 1, 27, 59, 3, 9, 1, 1, 1, 3, 15, 5, 1, 477, 1, 3, 7, 11, 1, 1, 2175, 17, 9, 7, 3, 747, 1, 41571, 1, 5, 19, 627, 13, 1, 1, 9, 5, 153
Offset: 1

Views

Author

Wolfdieter Lang, Oct 29 2015

Keywords

Comments

See A263008. E(n)*a(n)^2 - D(n)*A263008(n)^2 = +2, n >= 1, with the 2-happy couple (D(n), E(n)) = (A191856(n), A191857(n)). The 2-happy numbers D(n)*E(n) are given by A007970(n).
In the Zumkeller link "Initial Happy Factorization Data" given in A191860 the a(n) = U0(n) numbers appear for the t = 2 rows in column w.

Examples

			n = 4: 2-happy number A007970(4) = 11 = 1*11 =
  A191856(4)*A191857(4). 11*a(4)^2 - 1*A263008(4)^2 = 11*1^2 - 1*3^2 = +2. This is the smallest positive solution for given (D, E) = (1, 11).

Links

Crossrefs

Cf. A007970, A191856, A191857, A191860, A263008, A262026 , A262027, A262028.

Formula

A191857(n)*a(n)^2 - A191856(n)*A263008(n)^2 = +2, and A263008(n) with a(n) is the smallest positive
solution for the given 1-happy couple (A191856(n), A191857(n)).
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.