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.

A237606 Positive integers k such that x^2 - 8xy + y^2 + k = 0 has integer solutions.

Original entry on oeis.org

6, 11, 14, 15, 24, 35, 44, 51, 54, 56, 59, 60, 71, 86, 96, 99, 110, 119, 126, 131, 134, 135, 140, 150, 159, 176, 179, 191, 204, 206, 215, 216, 224, 231, 236, 239, 240, 251, 254, 275, 284, 294, 311, 315, 326, 335, 339, 344, 350, 359, 366, 371, 374, 375, 384
Offset: 1

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Author

Colin Barker, Feb 10 2014

Keywords

Comments

From Klaus Purath, Feb 17 2024: (Start)
Positive numbers of the form 15x^2 - y^2. The reduced form is -x^2 + 6xy + 6y^2.
Even powers of terms as well as products of an even number of terms belong to A243188. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (au^2 - cv^2)(ax^2 - cy^2) = (aux + cvy)^2 - ac(uy + vx)^2 and (au^2 + cv^2)(ax^2 + cy^2) = (aux - cvy)^2 + ac(uy + vx)^2 for all a, c, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
Odd powers of terms as well as products of an odd number of terms belong to the sequence. This can be proved with respect to the forms [a,0,-c] and [a, 0, +c] by the following identities: (as^2 - ct^2)(au^2 - cv^2)(ax^2 - cy^2) = a[s(aux + cvy) + ct(uy + vx)]^2 - c[as(uy + vx) + t(aux + cvy)]^2 and (as^2 + ct^2)(au^2 + cv^2)(ax^2 + cy^2) = a[s(aux - cvy) - ct(uy + vx)]^2 + c[as(uy + vx) + t(aux - cvy)]^2 for all a, c, s, t, u, v, x, y in R. This can be verified by expanding both sides of the equations. Generalization (conjecture): This multiplication rule applies to all sequences represented by any binary quadratic form [a, b, c].
If we denote any term of this sequence by B and correspondingly of A243189 by C and of A243190 by D, then B*C = D, C*D = B and B*D = C. This can be proved by the following identities, where the sequence (B) is represented by [kn, 0, -1], (C) by [n, 0, -k] and (D) by [k, 0, -n].
Proof of B*C = D: (knu^2 - v^2)(nx^2 - ky^2) = k(nux + vy)^2 - n(kuy + vx)^2 for k, n, u, v, x, y in R.
Proof of C*D = B: (nu^2 - kv^2)(kx^2 - ny^2) = kn(ux + vy)^2 - (nuy + kvx)^2 for k, n, u, v, x, y in R.
Proof of B*D = C: (knu^2 - v^2)(kx^2 - ny^2) = n(kux + vy)^2 - k(nuy + vx)^2 for k, n, u, v, x, y in R. This can be verified by expanding both sides of the equations.
Generalization (conjecture): If there are three sequences of a given positive discriminant that are represented by the forms [a1, b1, c1], [a2, b2, c2] and [a1*a2, b3, c3] for a1, a2 != 1, then the BCD rules apply to these sequences. (End)

Examples

			6 is in the sequence because x^2 - 8xy + y^2 + 6 = 0 has integer solutions, for example (x, y) = (1, 7).

Links

Crossrefs

Cf. A070997 (k = 6), A199336 (k = 14), A001091 (k = 15), A077248 (k = 35).
Cf. A031363, A084917, A237351, A237599, A237609, A237610.
For primes see A141302.
Cf. A378710, A378711 (subsequence of properly represented numbers and fundamental solutions).

A237262 Values of x in the solutions to x^2 - 8*x*y + y^2 + 11 = 0, where 0 < x < y.

Original entry on oeis.org

1, 2, 6, 15, 47, 118, 370, 929, 2913, 7314, 22934, 57583, 180559, 453350, 1421538, 3569217, 11191745, 28100386, 88112422, 221233871, 693707631, 1741770582, 5461548626, 13712930785, 42998681377, 107961675698, 338527902390, 849980474799, 2665224537743
Offset: 1

Views

Author

Colin Barker, Feb 05 2014

Keywords

Comments

The corresponding values of y are given by a(n+2).
Also values of y in the solutions to the negative Pell equation x^2 - 15*y^2 = -11. - Colin Barker, Jan 25 2017

Examples

			6 is a term because (x, y) = (6, 47) is a solution to x^2 - 8xy + y^2 + 11 = 0.

Links

Crossrefs

Cf. A001091, A070997, A199336, A281584.
For first and second differences see A322780, A199336.

Programs

  • Mathematica
    LinearRecurrence[{0,8,0,-1},{1,2,6,15},30] (* Harvey P. Dale, Sep 06 2020 *)
  • PARI
    Vec(-x*(x-1)*(x^2+3*x+1)/(x^4-8*x^2+1) + O(x^100))

Formula

G.f.: -x*(x-1)*(x^2 + 3*x + 1) / (x^4 - 8*x^2 + 1).
a(n) = 8*a(n-2) - a(n-4) for n > 4.
a(n) = (11*a(n-1) - 4*a(n-2))/3 if n is odd; a(n) = (11*a(n-1) - 3*a(n-2))/4 if n is even. - R. J. Mathar, Jun 18 2014

A322780 First differences of A237262.

Original entry on oeis.org

1, 4, 9, 32, 71, 252, 559, 1984, 4401, 15620, 34649, 122976, 272791, 968188, 2147679, 7622528, 16908641, 60012036, 133121449, 472473760, 1048062951, 3719778044, 8251382159, 29285750592, 64962994321, 230566226692, 511452572409, 1815244062944
Offset: 1

Views

Author

N. J. A. Sloane, Jan 05 2019, following a suggestion from Stan Wagon

Keywords

Links

Crossrefs

Cf. A237262, A199336.

Programs

  • PARI
    Vec(x*(1 + 4*x + x^2) / (1 - 8*x^2 + x^4) + O(x^40)) \\ Colin Barker, Jan 06 2019

Formula

From Colin Barker, Jan 06 2019: (Start)
G.f.: x*(1 + 4*x + x^2) / (1 - 8*x^2 + x^4).
a(n) = 8*a(n-2) - a(n-4) for n>4.
(End)

A199338 y-values in the solution to 15*x^2 - 14 = y^2.

Original entry on oeis.org

1, 11, 19, 89, 151, 701, 1189, 5519, 9361, 43451, 73699, 342089, 580231, 2693261, 4568149, 21203999, 35964961, 166938731, 283151539, 1314305849, 2229247351, 10347508061, 17550827269, 81465758639, 138177370801, 641378561051, 1087868139139, 5049562729769
Offset: 1

Views

Author

Sture Sjöstedt, Nov 08 2011

Keywords

Comments

When are both n+1 and 15*n+1 perfect squares? This problem gives the equation 15*x^2-14=y^2.
Essentially the same as A103201. - R. J. Mathar, Nov 11 2011

Links

Crossrefs

Cf. A199336.

Programs

  • Magma
    m:=29; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(x*(1+x)*(1+10*x+x^2)/(1-8*x^2+x^4))); // Bruno Berselli, Nov 08 2011
  • Mathematica
    LinearRecurrence[{0, 8, 0, -1}, {1, 11, 19, 89}, 50]

Formula

a(n+4) = 8*a(n+2) - a(n) with a(1)=1, a(2)=11, a(3)=19, a(4)=89.
G.f.: x*(1+x)*(1+10*x+x^2)/(1-8*x^2+x^4). - Bruno Berselli, Nov 08 2011
Showing 1-4 of 4 results.

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

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