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.

A129625 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.

Original entry on oeis.org

0, 75, 432, 699, 1092, 3115, 4660, 6943, 18724, 27727, 41032, 109695, 162168, 239715, 639912, 945747, 1397724, 3730243, 5512780, 8147095, 21742012, 32131399, 47485312, 126722295, 187276080, 276765243, 738592224, 1091525547, 1613106612
Offset: 1

Views

Author

Mohamed Bouhamida, May 30 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+233, y).
Corresponding values y of solutions (x, y) are in A157297.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A157297, A001652, A129288, A129289, A129298, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

Programs

  • Magma
    I:=[0,75,432,699,1092,3115,4660]; [n le 7 select I[n] else Self(n-1) + 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..30]]; // G. C. Greubel, Mar 29 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,75,432,699,1092,3115,4660}, 50] (* G. C. Greubel, Mar 29 2018 *)
  • PARI
    {forstep(n=0, 1700000000, [3, 1], if(issquare(2*n^2+466*n+54289), print1(n, ",")))};

Formula

a(n) = 6*a(n-3) -a(n-6) +466 for n > 6; a(1)=0, a(2)=75, a(3)=432, a(4)=699, a(5)=1092, a(6)=3115.
G.f.: x*(75 +357*x +267*x^2 -57*x^3 -119*x^4 -57*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 233*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 11 2009

A157297 Positive numbers y such that y^2 is of the form x^2+(x+233)^2 with integer x.

Original entry on oeis.org

185, 233, 317, 793, 1165, 1717, 4573, 6757, 9985, 26645, 39377, 58193, 155297, 229505, 339173, 905137, 1337653, 1976845, 5275525, 7796413, 11521897, 30748013, 45440825, 67154537, 179212553, 264848537, 391405325, 1044527305, 1543650397
Offset: 1

Views

Author

Klaus Brockhaus, Apr 11 2009

Keywords

Comments

(-57, a(1)) and (A129625(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+233)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (82611+44030*sqrt(2))/233^2 for n mod 3 = 1.

Examples

			(-57, a(1)) = (-57, 185) is a solution: (-57)^2+(-57+233)^2 = 3249+30976 = 34225 = 185^2.
(A129625(1), a(2)) = (0, 233) is a solution: 0^2+(0+233)^2 = 54289 = 233^2.
(A129625(3), a(4)) = (432, 793) is a solution: 432^2+(432+233)^2 = 186624+442225 = 628849 = 793^2.

Links

Crossrefs

Cf. A129625, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157298 (decimal expansion of (251+66*sqrt(2))/233), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

Programs

  • Magma
    I:=[185,233,317,793,1165,1717]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Mar 29 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1}, {185,233,317,793,1165,1717}, 50] (* G. C. Greubel, Mar 29 2018 *)
  • PARI
    {forstep(n=-60, 1100000000, [3,1], if(issquare(2*n^2+466*n+54289, &k),print1(k, ",")))};

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=185, a(2)=233, a(3)=317, a(4)=793, a(5)=1165, a(6)=1717.
G.f.: (1-x)*(185 +418*x +735*x^2 +418*x^3 +185*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 233*A001653(k) for k >= 1.

A157298 Decimal expansion of (251+66*sqrt(2))/233.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 11 2009

Keywords

Comments

lim_{n -> infinity} b(n)/b(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {1, 2}, b = A129625.
lim_{n -> infinity} b(n)/b(n-1) = (251+66*sqrt(2))/233 for n mod 3 = {0, 2}, b = A157297.

Examples

			(251+66*sqrt(2))/233 = 1.47784590178808700953...

Links

Crossrefs

Cf. A129625, A157297, A002193 (decimal expansion of sqrt(2)), A157299 (decimal expansion of (82611+44030*sqrt(2))/233^2).

Programs

  • Magma
    (251+66*Sqrt(2))/233; // G. C. Greubel, Mar 29 2018
  • Mathematica
    RealDigits[(251 + 66*Sqrt[2])/233, 10, 100][[1]] (* G. C. Greubel, Mar 29 2018 *)
  • PARI
    (251+66*sqrt(2))/233 \\ G. C. Greubel, Mar 29 2018

Formula

(251+66*sqrt(2))/233 = (22+3*sqrt(2))/(22-3*sqrt(2)).
Showing 1-3 of 3 results.

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