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.

A097742 Pell equation solutions (10*b(n))^2 - 101*a(n)^2 = -1 with b(n)=A097741(n), n >= 0.

Original entry on oeis.org

1, 401, 161201, 64802401, 26050404001, 10472197606001, 4209797387208401, 1692328077460171201, 680311677341601614401, 273483601963246388818001, 109939727677547706703222001, 44195497042772214848306426401, 17766479871466752821312480191201, 7142080712832591861952768730436401
Offset: 0

Views

Author

Wolfdieter Lang, Aug 31 2004

Keywords

Examples

			(x,y) = (10*1=10;1), (4030=10*403;401), (1620050=10*162005;161201), ... give the positive integer solutions to x^2 - 101*y^2 =-1.

Links

Crossrefs

Cf. A097740 for S(n, 402).
Row 10 of array A188647.

Programs

  • GAP
    a:=[1,401];; for n in [3..20] do a[n]:=402*a[n-1]-a[n-2]; od; a; # G. C. Greubel, Aug 01 2019
  • Magma
    I:=[1,401]; [n le 2 select I[n] else 402*Self(n-1) - Self(n-2): n in [1..20]]; // G. C. Greubel, Aug 01 2019
  • Mathematica
    LinearRecurrence[{402, -1},{1, 401},12] (* Ray Chandler, Aug 12 2015 *)
  • PARI
    my(x='x+O('x^20)); Vec((1-x)/(1-402*x+x^2)) \\ G. C. Greubel, Aug 01 2019
  • Sage
    ((1-x)/(1-402*x+x^2)).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Aug 01 2019

Formula

a(n) = S(n, 2*201) - S(n-1, 2*201) = T(2*n+1, sqrt(101))/sqrt(101), with Chebyshev polynomials of the 2nd and first kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x); and A053120 for the T-triangle.
a(n) = ((-1)^n)*S(2*n, 20*i) with the imaginary unit i and Chebyshev polynomials S(n, x) with coefficients shown in A049310.
G.f.: (1-x)/(1-402*x+x^2).
a(n) = 402*a(n-1) - a(n-2), n > 1; a(0)=1, a(1)=401. - Philippe Deléham, Nov 18 2008

A097741 Pell equation solutions (10*a(n))^2 - 101*b(n)^2 = -1 with b(n) = A097742(n), n >= 0.

Original entry on oeis.org

1, 403, 162005, 65125607, 26180332009, 10524428342011, 4230794013156413, 1700768668860536015, 683704774087922321617, 274847618414675912754019, 110488058897925629004794021, 44415924829347688184014442423, 17855091293338872724344801060025
Offset: 0

Views

Author

Wolfdieter Lang, Aug 31 2004

Keywords

Examples

			(x,y) = (10*1=10;1), (4030=10*403;401), (1620050=10*162005;161201), ... give the positive integer solutions to x^2 - 101*y^2 =-1.

Links

Crossrefs

Cf. A097740 for S(n, 2*201).
Cf. similar sequences of the type (1/k)*sinh((2*n+1)*arcsinh(k)) listed in A097775.

Programs

  • Mathematica
    LinearRecurrence[{402, -1}, {1, 403}, 20] (* or *) CoefficientList[Series[(1 + x)/(1 - 402 x + x^2), {x, 0, 20}], x]  (* Harvey P. Dale, Apr 20 2011 *)
a[n_] := Floor[(10 + Sqrt[101])^(2 n + 1)]/20;
Table[a[n], {n, 0, 11}] (* Peter Luschny, Apr 05 2018 *)
  • PARI
    x='x+O('x^99); Vec((1+x)/(1-2*201*x+x^2)) \\ Altug Alkan, Apr 05 2018

Formula

G.f.: (1 + x)/(1 - 2*201*x + x^2).
a(n) = S(n, 2*201) + S(n-1, 2*201) = S(2*n, 2*sqrt(101)), with Chebyshev polynomials of the 2nd kind. See A049310 for the triangle of S(n, x)= U(n, x/2) coefficients. S(-1, x) := 0 =: U(-1, x).
a(n) = ((-1)^n)*T(2*n+1, 10*i)/(10*i) with the imaginary unit i and Chebyshev polynomials of the first kind. See the T-triangle A053120.
a(n) = 402*a(n-1) - a(n-2) for n > 1, a(0)=1, a(1)=403. - Philippe Deléham, Nov 18 2008
a(n) = (1/10)*sinh((2*n + 1)*arcsinh(10)). - Bruno Berselli, Apr 03 2018
Let h = (10 + sqrt(101))^(2*n+1) then a(n) = (h-1/h)/20 and a(n) = floor(h/20). - Peter Luschny, Apr 05 2018

A227152 Nonnegative solutions of the Pell equation x^2 - 101*y^2 = +1. Solutions x = a(n).

Original entry on oeis.org

1, 201, 80801, 32481801, 13057603201, 5249124005001, 2110134792407201, 848268937423689801, 341002002709530892801, 137081956820293995216201, 55106605639755476546020001, 22152718385224881277504824201
Offset: 0

Views

Author

Wolfdieter Lang, Jul 05 2013

Keywords

Comments

The Pell equation x^2 - 101*y^2 = +1 has only proper solutions, namely x(n) = a(n) and y(n) = 20*A097740(n), n>= 0.

Examples

			n=0: 1^2 - 101*0^2  = +1 (a proper, but not a positive solution),
n=1: 201^2 - 101*(20*1)^2 = +1, where 20  is the positive fundamental y-solution.
n=2: 80801^2 - 101*(20*402)^2 = +1, where 80801 = 7^2*17*97  and 20*402 = 8040 = 2^3*3*5*67.

References

  • T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. VI, 56., pp. 115-200.
  • O. Perron, Die Lehre von den Kettenbruechen, Band I, Teubner, Stuttgart, 1954, Paragraph 27, pp. 92-95.

Links

Crossrefs

Cf. A097740 (y/20 solutions and S(n,402)), A049310, A053120.

Programs

  • Mathematica
    LinearRecurrence[{402,-1},{1,201},20] (* Harvey P. Dale, Jan 17 2020 *)

Formula

a(n) = (S(n, 2*201) - S(n-2, 2*201))/2 = T(n, 201) with the Chebyshev S- and T-polynomials (see A049310 and A053120, respectively). S(n, -2) = -1, S(n, -1) = 0. For S(n, 2*201) see A097740.
a(n) = 2*201*a(n-1) - a(n-2), n >= 1, with input a(-1) = 201 and a(0) = 1.
O.g.f.: (1 - 201*x)/(1 - 2*201*x + x^2).
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.