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.

A262025 a(n) = (A262024(n)-1)/2: a(n)*(a(n) + 1) = d(n)*Y(n)^2 with d(n) = A007969 and Y(n) = A261250(n).

Original entry on oeis.org

1, 4, 2, 9, 3, 324, 7, 16, 8, 4, 27, 98, 25, 63, 4900, 5, 11, 17, 36, 18, 12, 1024, 6, 99, 80, 12167, 49, 324, 33124, 242, 44, 7, 75, 9801, 15, 883159524, 31, 64, 32, 16, 3887, 125, 8, 1140624, 1849, 28899, 175, 26, 81, 27, 142884, 5202, 250000, 9, 575, 6075, 1071647, 19, 31404816, 49, 100, 50, 20, 16040025, 675, 79035335993124, 10, 147, 63, 602176, 512, 4900, 324, 153458
Offset: 1

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Author

Wolfdieter Lang, Sep 19 2015

Keywords

Comments

The positive fundamental solutions (x0(n), y0(n)) of the Pell equation x^2 - d(n) y^2 = +1, with d not a square, have only even y solutions for d(n) = A007969 (Conway's products of 1-happy couples). The proof is now given in the W. Lang link under A007969. The solutions x0 and y0 = 2*Y0 are given in A262024 and 2*A261250, respectively. The numbers X0(n) = (x0(n) - 1)/2 = a(n) satisfy a(n)*(a(n) + 1) = d(n)*Y0(n)^2. See the mentioned link.

Crossrefs

Cf. A007969, A007970, A262024, A261250.

A261250 One half of the even entries of A033317.

Original entry on oeis.org

1, 2, 1, 3, 1, 90, 2, 4, 2, 1, 6, 21, 5, 12, 910, 1, 2, 3, 6, 3, 2, 160, 1, 15, 12, 1794, 7, 45, 4550, 33, 6, 1, 10, 1287, 2, 113076990, 4, 8, 4, 2, 468, 15, 1, 133500, 215, 3315, 20, 3, 9, 3, 15498, 561, 26500, 1, 60, 630, 110532, 2, 3188676, 5, 10, 5, 2, 1557945, 65, 7570212227550, 1, 14, 6, 56648, 48, 455, 30, 14127
Offset: 1

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Author

Wolfdieter Lang, Sep 16 2015

Keywords

Comments

2*a(n) = y0(n) is the positive fundamental solution satisfying the Pell equation x0(n)^2 + D(n)*y0(n)^2 = +1 with D(n) coinciding apparently with Conway's rectangular numbers r(n) = A007969(n). The corresponding x0 values are given in A262024.
For a proof of this coincidence see the W. Lang link under A007969. - Wolfdieter Lang, Oct 04 2015

Examples

			The [r(n), x0(n), y0(n)] values for n = 1..16 are:
[2, 3, 2], [5, 9, 4], [6, 5, 2], [10, 19, 6],
[12, 7, 2], [13, 649, 180], [14, 15, 4],
[17, 33, 8], [18, 17, 4], [20, 9, 2],
[21, 55, 12], [22, 197, 42], [26, 51, 10],
[28, 127, 24], [29, 9801, 1820], [30, 11, 2], ...

Crossrefs

Cf. A033317, A033313, A000037, A007969, A262024, A262025.

Programs

  • Mathematica
    PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[Last[cf]]; If[n == 0, Return[{}]]; If[OddQ[n], n = 2 n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}];
Select[DeleteCases[PellSolve /@ Range[200], {}][[All, 2]], EvenQ]/2 (* Jean-François Alcover, Aug 12 2023, using the PellSolve code given in A033317 *)

A261249 Number of classes of proper solutions of the Pell equation x^2 - D(n) y^2 = +4 for D(n) = A079896(n), n >= 1.

Original entry on oeis.org

2, 0, 1, 2, 0, 0, 2, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 0, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 2, 1, 0, 0, 2, 0, 0, 1, 0, 0, 0
Offset: 1

Views

Author

Wolfdieter Lang, Sep 16 2015

Keywords

Comments

See the W. Lang link on A225953, Table 2. References will also be found there. For the present class number see especially Theorem 109 pp. 207-208 of the Nagell reference.
These class numbers should not be confused with the class numbers of indefinite binary quadratic forms of discriminant D(n), which are given in A087048(n).
If a(n) = 2 then the proper positive fundamental solution for the second class [x2(n), y2(n)] is obtained from the solution of the first class [x1(n), y1(n)] (shown in the mentioned Table 2 under Pell(X, Y)) by application of the matrix M(n) = [[x0(n), D(n)*y0(n)], [y0(n), x0(n)]] on (x1(n), -y1(n))^T (T for transposed), where x0(n) and y0(n) is the positive (proper) fundamental solution of x^2 - D(n)*y^2 = +1 found under A033313 and A033317 for the appropriate D from A000037. Application of positive powers of M(n) to the proper positive fundamental solution of each class produces all positive solutions.
If a(n) = 1 the class is called ambiguous (see Nagell, p. 205). In this case the proper positive fundamental solution [x1(n), y1(n)] = [x(n), y(n)] and the negative one [x1(n), -y1(n)] belong to the same class.
For every D(n) = A079896(n) there is the improper positive fundamental solution [2*x0(n), 2*y0(n)].
Conjecture: For even D(n), i.e., D from 4*A000037, and a(n) = 0 one finds for r(n) = D(n)/4 coincidence with Conway's so-called rectangular numbers A007969. The first D values are 8, 20, 24, 40, 48, 52, 56, 68, 72, 80, ... This is equivalent to the conjecture that X^2 - r*y^2 = +1 has an even fundamental positive solution y = y0 precisely for the numbers A007969 (because x has to be even, x = 2*X, and whenever y0 is even all y solutions are even). See A261250 and A262024 for the y0 and x0 values, respectively.

Examples

			n=1: D(1) = 5 = A000037(3) with the a(1) = 2 proper positive fundamental solutions [x, y] = [3, 1] and [7, 3] for the two classes.
  [x0(1), y0(1)] = [A033313(3), A033317(3)] = [9, 4], and (7, 3)^T = [[9, 4*5], [4, 9]] (3, -1)^T.
  All other positive solutions in each of the two classes are obtained by applying positive powers of this matrix M(5) to the fundamental solutions.
  The improper positive fundamental solution is [2*9, 2*4] = [18, 8].
n=2: D(2) = 8 = A000037(6) has a(2) = 0, hence there are only the improper solutions obtainable from [2*3, 2*1] = [6, 2], the smallest positive one. For this even D one has, with x = 2*X, X^2 - 8/4 y^2 = +1, which has an even positive fundamental solution y0 = 2, and r(2) = D(2)/4 = 2 is A007969(1).

References

  • Nagell, T. Introduction to number theory, Chelsea Publishing Company, 1964, page 52.

Crossrefs

Cf. A079896, A225953, A087048, A033313, A033317, A261250, A262024.

Extensions

Offset corrected by Robin Visser, Jun 08 2025
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