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.

A129857 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.

Original entry on oeis.org

0, 235, 1696, 2571, 3796, 12075, 17140, 24255, 72468, 101983, 143448, 424447, 596472, 838147, 2475928, 3478563, 4887148, 14432835, 20276620, 28486455, 84122796, 118182871, 166033296, 490305655, 688822320, 967715035, 2857712848
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+857, y).
Corresponding values y of solutions (x, y) are in A160206.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160206, A001652, A123654, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat  Coefficients(R!(x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5)/((1-x)*(1-6*x^3+x^6))) );  // G. C. Greubel, May 03 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,235,1696,2571,3796,12075, 17140}, 30] (* or *) CoefficientList[Series[x (235+1461x+875x^2-185x^3- 487x^4- 185x^5)/((1-x)(1-6x^3+x^6)),{x,0,30}],x] (* Harvey P. Dale, Apr 24 2011 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1714*n+734449), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1714 for n > 6; a(1)=0, a(2)=235, a(3)=1696, a(4)=2571, a(5)=3796, a(6)=12075.
G.f.: x*(235+1461*x+875*x^2-185*x^3-487*x^4-185*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 857*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160206 Positive numbers y such that y^2 is of the form x^2+(x+857)^2 with integer x.

Original entry on oeis.org

697, 857, 1117, 3065, 4285, 6005, 17693, 24853, 34913, 103093, 144833, 203473, 600865, 844145, 1185925, 3502097, 4920037, 6912077, 20411717, 28676077, 40286537, 118968205, 167136425, 234807145, 693397513, 974142473, 1368556333
Offset: 1

Views

Author

Klaus Brockhaus, May 18 2009

Keywords

Comments

(-185, a(1)) and (A129857(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+857)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (907+210*sqrt(2))/857 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1208787+678878*sqrt(2))/857^2 for n mod 3 = 1.

Examples

			(-185, a(1)) = (-185, 697) is a solution: (-185)^2+(-185+857)^2 = 34225+451584 = 485809 = 697^2.
(A129857(1), a(2)) = (0, 857) is a solution: 0^2+(0+857)^2 = 734449 = 857^2.
(A129857(3), a(4)) = (1696, 3065) is a solution: 1696^2+(1696+857)^2 = 2876416+6517809 = 9394225 = 3065^2.

Links

Crossrefs

Cf. A129857, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857), A160208 (decimal expansion of (1208787+678878*sqrt(2))/857^2).

Programs

  • Magma
    I:=[697,857,1117,3065,4285,6005]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..40]]; // G. C. Greubel, May 14 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1}, {697,857,1117,3065,4285,6005}, 50] (* G. C. Greubel, May 14 2018 *)
  • PARI
    {forstep(n=-188, 10000000, [3, 1], if(issquare(2*n^2 +1714*n +734449, &k), print1(k, ",")))}
  • PARI
    x='x+O('x^30); Vec((1-x)*(697+1554*x+2671*x^2+1554*x^3 +697*x^4 )/(1-6*x^3+x^6)) \\ G. C. Greubel, May 14 2018

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=697, a(2)=857, a(3)=1117, a(4)=3065, a(5)=4285, a(6)=6005.
G.f.: (1-x)*(697+1554*x+2671*x^2+1554*x^3+697*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 857*A001653(k) for k >= 1.

A160208 Decimal expansion of (1208787 +678878*sqrt(2))/857^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 18 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129857.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160206.

Examples

			(1208787 +678878*sqrt(2))/857^2 = 2.95305116461009821041...

Links

Crossrefs

Cf. A129857, A160206, A002193 (decimal expansion of sqrt(2)), A160207 (decimal expansion of (907+210*sqrt(2))/857).

Programs

  • Magma
    (1208787 +678878*Sqrt(2))/857^2; // G. C. Greubel, Apr 08 2018
  • Mathematica
    RealDigits[(1208787+678878Sqrt[2])/857^2,10,120][[1]] (* Harvey P. Dale, Nov 09 2012 *)
  • PARI
    (1208787 +678878*sqrt(2))/857^2 \\ G. C. Greubel, Apr 08 2018

Formula

Equals (1394 +487*sqrt(2))/(1394 -487*sqrt(2));
Equals (3+2*sqrt(2))*(42-5*sqrt(2))^2/(42+5*sqrt(2))^2.
Showing 1-3 of 3 results.

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