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.

A129725 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+521)^2 = y^2.

Original entry on oeis.org

0, 100, 1159, 1563, 2079, 8080, 10420, 13416, 48363, 61999, 79459, 283140, 362616, 464380, 1651519, 2114739, 2707863, 9627016, 12326860, 15783840, 56111619, 71847463, 91996219, 327043740, 418758960, 536194516, 1906151863, 2440707339, 3125171919, 11109868480
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 02 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+521, y).
Corresponding values y of solutions (x, y) are in A160583.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (537+92*sqrt(2))/521 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (520659+314170*sqrt(2))/521^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160583, A001652, A129642, A156035 (decimal expansion of 3+2*sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 100, 1159, 1563, 2079, 8080, 10420}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 13 2012 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1042*n+271441), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1042 for n > 6; a(1)=0, a(2)=100, a(3)=1159, a(4)=1563, a(5)=2079, a(6)=8080.
G.f.: x*(100+1059*x+404*x^2-84*x^3-353*x^4-84*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 521*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Jun 08 2009

A160584 Decimal expansion of (537 +92*sqrt(2))/521.

Original entry on oeis.org

1, 2, 8, 0, 4, 3, 6, 9, 4, 3, 8, 3, 5, 5, 5, 6, 1, 3, 1, 4, 5, 8, 2, 6, 3, 6, 5, 1, 8, 7, 5, 8, 0, 0, 8, 1, 2, 3, 3, 8, 5, 9, 8, 4, 8, 8, 5, 5, 0, 2, 4, 8, 0, 2, 8, 2, 7, 6, 8, 8, 0, 1, 0, 7, 2, 8, 4, 3, 5, 1, 9, 9, 2, 3, 5, 7, 9, 9, 2, 0, 2, 9, 8, 9, 2, 9, 6, 6, 7, 0, 4, 7, 1, 8, 1, 2, 4, 8, 0, 7, 4, 6, 9, 5, 2
Offset: 1

Views

Author

Klaus Brockhaus, Jun 08 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129725.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160583.

Examples

			(537 +92*sqrt(2))/521 = 1.28043694383555613145...

Links

Crossrefs

Cf. A129725, A160583, A002193 (decimal expansion of sqrt(2)), A160585 (decimal expansion of (520659+314170*sqrt(2))/521^2).

Programs

  • Magma
    (537 +92*Sqrt(2))/521; // G. C. Greubel, Apr 07 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=150:
RealDigits(evalf((537+92*sqrt(2))/521))[1]; # Muniru A Asiru, Apr 08 2018
  • Mathematica
    RealDigits[(537 +92*Sqrt[2])/521, 10, 100][[1]] (* G. C. Greubel, Apr 07 2018 *)
  • PARI
    (537 +92*sqrt(2))/521 \\ G. C. Greubel, Apr 07 2018

Formula

Equals (23 +2*sqrt(2))/(23 -2*sqrt(2)).

A160585 Decimal expansion of (520659 +314170*sqrt(2))/521^2.

Original entry on oeis.org

3, 5, 5, 4, 9, 6, 2, 1, 2, 7, 6, 4, 7, 4, 6, 3, 9, 8, 4, 7, 4, 0, 7, 9, 6, 5, 1, 3, 7, 3, 5, 8, 0, 5, 7, 3, 8, 0, 5, 8, 1, 1, 8, 7, 5, 5, 4, 0, 9, 7, 4, 1, 9, 9, 2, 3, 6, 6, 3, 6, 1, 2, 8, 3, 9, 7, 9, 0, 1, 7, 2, 5, 5, 1, 6, 2, 0, 4, 3, 2, 0, 0, 5, 3, 2, 0, 7, 2, 3, 5, 3, 3, 2, 8, 0, 3, 5, 0, 1, 6, 4, 7, 1, 9, 3
Offset: 1

Views

Author

Klaus Brockhaus, Jun 08 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129725.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160583.

Examples

			(520659 +314170*sqrt(2))/521^2 = 3.55496212764746398474...

Links

Crossrefs

Cf. A129725, A160583, A002193 (decimal expansion of sqrt(2)), A160584 (decimal expansion of (537+92*sqrt(2))/521).

Programs

  • Magma
    (520659 +314170*Sqrt(2))/521^2; // G. C. Greubel, Apr 07 2018
  • Maple
    with(MmaTranslator[Mma]): Digits:=150:
RealDigits(evalf((520659+314170*sqrt(2))/521^2))[1]; # Muniru A Asiru, Apr 08 2018
  • Mathematica
    RealDigits[(520659+314170Sqrt[2])/521^2,10,120][[1]] (* Harvey P. Dale, Aug 23 2011 *)
  • PARI
    (520659 +314170*sqrt(2))/521^2 \\ G. C. Greubel, Apr 07 2018

Formula

Equals (890 +353*sqrt(2))/(890 -353*sqrt(2)).
Equals (3 +2*sqrt(2))*(23 -2*sqrt(2))^2/(23 +2*sqrt(2))^2.
Showing 1-3 of 3 results.

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