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.

A118675 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.

Original entry on oeis.org

0, 16, 85, 141, 225, 616, 940, 1428, 3705, 5593, 8437, 21708, 32712, 49288, 126637, 190773, 287385, 738208, 1112020, 1675116, 4302705, 6481441, 9763405, 25078116, 37776720, 56905408, 146166085, 220178973, 331669137, 851918488, 1283297212, 1933109508
Offset: 1

Views

Author

Mohamed Bouhamida, May 19 2006

Keywords

Comments

Also values x of Pythagorean triples (x, x+47, y).
Corresponding values y of solutions (x, y) are in A159750.
For the generic case x^2+(x+p)^2 = y^2 with p = m^2-2 a (prime) number > 7 in A028871, see A118337.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159750, A028871, A118337, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

Programs

  • Magma
    m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3 +x^6)))); // G. C. Greubel, May 07 2018
  • Mathematica
    Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+47)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,16,85,141,225,616,940},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)
  • PARI
    {forstep(n=0, 100000000, [1, 3], if(issquare(2*n^2+94+2209), print1(n, ",")))}

Formula

a(n) = 6*a(n-3) -a(n-6) +94 for n > 6; a(1)=0, a(2)=16, a(3)=85, a(4)=141, a(5)=225, a(6)=616.
G.f.: x*(16+69*x+56*x^2-12*x^3-23*x^4-12*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 47*A001652(k) for k >= 0.

Extensions

Edited by Klaus Brockhaus, Apr 30 2009

A159750 Positive numbers y such that y^2 is of the form x^2+(x+47)^2 with integer x.

Original entry on oeis.org

37, 47, 65, 157, 235, 353, 905, 1363, 2053, 5273, 7943, 11965, 30733, 46295, 69737, 179125, 269827, 406457, 1044017, 1572667, 2369005, 6084977, 9166175, 13807573, 35465845, 53424383, 80476433, 206710093, 311380123, 469051025, 1204794713
Offset: 1

Views

Author

Klaus Brockhaus, Apr 30 2009

Keywords

Comments

(-12, a(1)) and (A118675(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+47)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (51+14*sqrt(2))/47 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3267+1702*sqrt(2))/47^2 for n mod 3 = 1.
For the generic case x^2+(x+p)^2=y^2 with p= m^2 -2 a prime number in A028871, m>=2, the x values are given by the sequence defined by: a(n)= 6*a(n-3) -a(n-6) +2*p with a(1)=0, a(2)= 2*m +2, a(3)= 3*m^2 -10*m +8, a(4)= 3*p, a(5)= 3*m^2 +10*m +8, a(6)= 20*m^2 -58*m +42. Y values are given by the sequence defined by: b(n)= 6*b(n-3) -b(n-6) with b(1)=p, b(2)= m^2 +2*m +2, b(3)= 5*m^2 -14*m +10, b(4)= 5*p, b(5)= 5*m^2 +14*m +10, b(6)= 29*m^2 -82*m +58. - Mohamed Bouhamida, Sep 09 2009

Examples

			(-12, a(1)) = (-12, 37) is a solution: (-12)^2+(-12+47)^2 = 144+1225 = 1369 = 37^2.
(A118675(1), a(2)) = (0, 47) is a solution: 0^2+(0+47)^2 = 2209 = 47^2.
(A118675(3), a(4)) = (85, 157) is a solution: 85^2+(85+47)^2 = 7225+17424 = 24649 = 157^2.

Links

Crossrefs

Cf. A118675, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47), A159752 (decimal expansion of (3267+1702*sqrt(2))/47^2).

Programs

  • Magma
    I:=[37,47,65,157,235,353]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 22 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1}, {37,47,65,157,235,353}, 50] (* G. C. Greubel, May 22 2018 *)
  • PARI
    {forstep(n=-12, 100000000, [1, 3], if(issquare(2*n^2+94*n+2209, &k), print1(k, ",")))};
  • PARI
    x='x+O('x^30); Vec((1-x)*(37+84*x+149*x^2+84*x^3+37*x^4)/(1 -6*x^3 +x^6)) \\ G. C. Greubel, May 22 2018

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=37, a(2)=47, a(3)=65, a(4)=157, a(5)=235, a(6)=353.
G.f.: (1-x)*(37+84*x+149*x^2+84*x^3+37*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 47*A001653(k) for k >= 1.

A159752 Decimal expansion of (3267 + 1702*sqrt(2))/47^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 30 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A118675.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159750.

Examples

			(3267 + 1702*sqrt(2))/47^2 = 2.56857921374332628929...

Links

Crossrefs

Cf. A118675, A159750, A002193 (decimal expansion of sqrt(2)), A159751 (decimal expansion of (51+14*sqrt(2))/47).

Programs

  • Magma
    (3267 +1702*Sqrt(2))/47^2; // G. C. Greubel, May 22 2018
  • Mathematica
    RealDigits[(3267 +1702*Sqrt[2])/47^2, 10, 100][[1]] (* G. C. Greubel, May 22 2018 *)
  • PARI
    (3267 +1702*sqrt(2))/47^2 \\ G. C. Greubel, May 22 2018

Formula

Equals (74 + 23*sqrt(2))/(74 - 23*sqrt(2)).
Equals (3 + 2*sqrt(2))*(7 - sqrt(2))^2/(7 + sqrt(2))^2.
Showing 1-3 of 3 results.

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