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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.

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A307168 First class of all proper positive solutions x1(n) = a(n) of the Pell equation x^2 - 7*y^2 = 9.

Original entry on oeis.org

11, 172, 2741, 43684, 696203, 11095564, 176832821, 2818229572, 44914840331, 715819215724, 11408192611253, 181815262564324, 2897636008417931, 46180360872122572, 735988137945543221, 11729629846256568964, 186938089402159560203, 2979279800588296394284, 47481538720010582748341
Offset: 1

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Author

Wolfdieter Lang, Mar 27 2019

Keywords

Comments

The corresponding solutions y1(n) are given in A307169.
The (generalized) Pell equation x^2 - 7*y^2 = 9 has two proper classes of positive solutions (x1(n), y1(n)) and (x2(n), y2(n)), for n >= 1, where x2 = A307172, and y2 = A307173.
The improper class of nonnegative solutions is given by (xi(n) = 3*X(n), yi(n) = 3*Y(n)), with the nonnegative solutions of the Pell equation X^2 - 7*Y^2 = +1, given by X(n) = A001081(n) and Y(n) = A001080(n), for n >= 0.
The proper positive solutions (x1(n), y1(n)) are given in matrix notation by -R(0)*R(2)*Auto(n)*R^{-1}(6)*(1, 0)^T (T for transposed) with the R-matrix R(t) = Matrix([[0, -1],[1, t]]) and its inverse R^{-1}(t) = Matrix([t, 1],[-1, 0]) and the automorphic matrix Auto = Matrix([2, 9],[3, 14]). The matrix power Auto^n can be given in terms of Chebyshev S-polynomials S(n, x=16) from A077412 as Auto^n = Matrix([S(n, 16) - 14*S(n-1, 16), 9*S(n-1, 16)],[3*S(n-1, 16), S(n, 16) - 2*S(n-1, 16)]).
This results from the reduced principal binary quadratic form F_p = [1, 4, -3] of the non-reduced Pell form FPell = [1, 0, -7], and the primitive representative parallel form FPara1 = [9, 8, 1] for discriminant 4*7 = 28 and the representation of 9. These forms are then connected via equivalence transformations using R(t) matrices.

Examples

			The solutions (x1(n), y1(n)) begin: (11, 4), (172, 65), (2741, 1036), (43684, 16511), (696203, 263140), (11095564, 4193729), (176832821, 66836524), (2818229572, 1065190655), (44914840331, 16976213956), ...
The solutions (x2(n), y2(n)) begin: (4, 1), (53, 20), (844, 319), (13451, 5084), (214372, 81025), (3416501, 1291316), (54449644, 20580031), (867777803, 327989180), (13829995204, 5227246849), ...
The improper solutions (xi(n), yi(n)) begin: (3, 0), (24, 9), (381, 144), (6072, 2295), (96771, 36576), (1542264, 582921), (24579453, 9290160), (391728984, 148059639), (6243084291, 2359664064), ...

Links

Crossrefs

Cf. A001080, A001081, A077412, A307169, A307172, A307173.

Formula

a(n) = 4*S(n, 16) - 53*S(n-1, 16), for n >= 1, with S(n, 16) = A077412(n).
a(n) = sqrt(9 + 7*A307169(n)), n >= 1.
G.f.: x*(11 - 4*x)/(1 - 16*x + x^2).

A358946 Positive integers that are properly represented by each primitive binary quadratic form of discriminant 28 that is properly equivalent to the principal form [1, 4, -3].

Original entry on oeis.org

1, 2, 9, 18, 21, 29, 37, 42, 53, 57, 58, 74, 81, 93, 106, 109, 113, 114, 133, 137, 141, 149, 162, 177, 186, 189, 193, 197, 217, 218, 226, 233, 249, 261, 266, 274, 277, 281, 282, 298, 309, 317, 329, 333, 337, 354, 361, 373, 378, 386, 389, 393, 394, 401, 413, 417, 421, 434, 449, 457, 466, 477, 498, 501
Offset: 1

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Author

Wolfdieter Lang, Jan 10 2023

Keywords

Comments

This is a subsequence of A242662, excluding the primitive forms of discriminant 28 with only improper representations of k, like k = 4, 8, 16, 25, 32, ... .
An indefinite binary quadratic primitive form F = a*x^2 + b*x*y + c*y^2 (gcd(a, b, c) = 1) with discriminant Disc = b^2 - 4*a*c = 28 = 2^2*7 is denoted by [a, b, c], or in matrix notation by MF = Matrix([[a, b/2], [b/2, c]]). Hence F = X*MF*X^T (T for transposed), where X = (x, y). See the two links for details and references.
Properly equivalent forms F' and F are related by a matrix R of determinant +1 like MF' = R^T*MF*R, and X'^T = R^{-1}*X^T.
Each primitive form, properly equivalent to the reduced principal form F_p = [1, 4, -3] for Disc = 28 (used in A242662), represents the given nonnegative k = a(n) values (and only these) properly with X = (x, y) and gcd(x, y) = 1. Modulo an overall sign change in X one can choose x nonnegative.
There are 8 = A082174(8) primitive reduced forms of Disc = 28 leading to 2 = A087048(8) (class number) cycles each of period 4, namely the principal cycle CyP = [[1, 4, -3], [-3, 2, 2], [2, 2, -3], [-3, 4, 1]] and the one (with outer signs flipped) CyP' = [[-1, 4, 3], [3, 2, -2], [-2, 2, 3], [3, 4, -1]].
There are A358947(n) representative parallel primitive forms (rpapfs) of discriminant Disc = 28 for k = a(n). This gives the number of proper fundamental representations X = (x, y), with x >= 0, of each primitive form [a, b, c], properly equivalent to the principal form F_p of Disc = 28.
For the negative integers k properly represented by primitive forms [a, b, c] properly equivalent to the principal form of Disc = 28 see A359476. The corresponding number of fundamental proper representations is given in A359477.
This and the three related sequences originated from a proposal by Klaus Purath proving that the form FKP := [1, -2, -6] of Disc = 28 represents k = k(m) = m^2 - 7 = A028881(m), for m >= 3, with the two fundamental representations X1(m) = (m+1, 1) and X2(m) = (11*m - 29, 3*m - 8). This form FKP is properly equivalent to the principal form F_p with R = Matrix([[1, -3], [0, 1]]). Hence all k = a(n) are represented by the form FKP, and A028881 is a subsequence of the present one.

Examples

			k = 9 = a(3): F = FPell = [1, 0, -7] is properly equivalent to F_p = [1, 4, -3] by two so-called half-reduced right neighbor R(t)-transformations, with the matrix R = R(t) = Matrix([[0, -1], [1, t]]), first with t = 0 then with t = 2. For FPell representing k = 9 with x > 0 and y > 0 see X_1(9, i) = (A307168(i), A307169(i)) and X_2(9, i) = (A307172(i), A307173(i)), for i >= 0. There are also the representations with y -> -y arising from the opposite fundamental solutions.
The 2 = A358947(3) rpapfs are F1 = [9, 8, 1] and F2 = [9, 10, 2]. They lead by proper equivalence transformations to a form of the above given principal cycle CyP. F1 -> [1, 4, -3] = F_p with matrix R(6), and F2 -> [2, 2, -3] with R(3). See the FIGURE, p. 10, of the linked paper.
Besides the primitive forms FPell, F1, F2 and the four forms of CyP also F' = [-7, 0, 1], and all primitive and properly equivalent forms represent k = 9. See the mentioned FIGURE, where FPa1 = F1, FPa1 = F2, Fpa2' = F_p^{(2)} = [2, 2, -3] and FPa2'' = F_p^{(3)} = [-3, 4, 1].

Links

Crossrefs

Cf. A028881, A242662, A307168, A307169, A307172, A307173, A358947, A359476, A359477.

A307173 Second class of all proper positive solutions y2(n) = a(n) of the Pell equation x^2 - 7*y^2 = 9.

Original entry on oeis.org

1, 20, 319, 5084, 81025, 1291316, 20580031, 327989180, 5227246849, 83307960404, 1327700119615, 21159893953436, 337230603135361, 5374529756212340, 85655245496262079, 1365109398183980924, 21756095125447432705, 346732412608974942356, 5525962506618151644991
Offset: 1

Views

Author

Wolfdieter Lang, Mar 28 2019

Keywords

Comments

The corresponding x2 solutions are given in A307172.
See A307172 for details.

Links

Crossrefs

Cf. A077421, A307168, A307169, A307172.

Formula

G.f.: x*(1 + 4*x)/(1 - 16*x + x^2).
a(n) = -4*S(n, 16) + 65*S(n-1, 16) for n >= 1, with S(n,16) = A077412(n).
a(n) = sqrt((A307172(n)^2 - 9)/7) for n >= 1.
Showing 1-3 of 3 results.

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