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 1-4 of 4 results.

A262028 a(n) = (A262026(n) - 1)/2.

Original entry on oeis.org

0, 1, 0, 1, 0, 19, 2, 0, 2, 136, 1, 0, 1, 265, 3, 0, 3, 34, 0, 2983, 206, 1, 4, 0, 4, 1, 10, 82, 2, 0, 11209, 2, 46, 52, 5, 0, 5, 209887, 25, 463, 10, 1, 3289414, 0, 70317346, 1, 52, 28, 2509567, 6, 0, 6, 76, 7, 156595
Offset: 1

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Author

Wolfdieter Lang, Oct 04 2015

Keywords

Comments

This is the column Y_0 of the Table of a proof given as a W. Lang link under A007970.
(x0(n), y0(n) = 2*a(n) + 1) with x0(n) = A262067(n) are the fundamental solutions of the Pell equation x^2 - d*y^2 = +1 with odd y. The d values coincide with d = d(n) = A007970(n). For a proof see the mentioned link.

Examples

			For the first triples [d(n), x0(n), 2*a(n) + 1] see A262066.

Crossrefs

Cf. A006970, A262026, A262067.

Formula

A262067(n)^2 - A007970(n)*(2*a(n) + 1)^2 = +1, n >= 1.

A262027 The positive fundamental solutions x = x0(n) for the Pell equation x^2 - d*y^2 = +1 with odd y = y0(n). Then d coincides with d(n) = A007970(n).

Original entry on oeis.org

2, 8, 3, 10, 4, 170, 24, 5, 26, 1520, 17, 6, 19, 3482, 48, 7, 50, 530, 8, 48842, 3480, 26, 80, 9, 82, 28, 197, 1574, 49, 10, 227528, 51, 962, 1126, 120, 11, 122, 4730624, 577, 10610, 244, 35, 77563250, 12, 1728148040, 37, 1324, 721, 64080026, 168, 13, 170, 2024, 199, 4190210
Offset: 1

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Author

Wolfdieter Lang, Oct 04 2015

Keywords

Comments

The corresponding values y = y0(n) are given by A262026(n).
This is a proper subset of A033313 corresponding to D values from d(n) = A007970(n).
For the proof that d(n) = A007970(n), the products of Conway's 2-happy couples, see the W. Lang link under A007970.
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

			For the first [d(n), x0(n), y0(n)] see A262026.

Crossrefs

Cf. A007970, A033313, A262026.

Formula

a(n)^2 - d(n)*y0(n)^2 = +1 with y0(n) = A262026(n) and d(n) = A007970(n). (x0(n) = a(n), y0(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

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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 1-4 of 4 results.

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