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.

A130004 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.

Original entry on oeis.org

0, 31, 1204, 1347, 1504, 8151, 8980, 9891, 48600, 53431, 58740, 284347, 312504, 343447, 1658380, 1822491, 2002840, 9666831, 10623340, 11674491, 56343504, 61918447, 68045004, 328395091, 360888240, 396596431, 1914027940, 2103411891, 2311534480, 11155773447
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+449, y).
Corresponding values y of solutions (x, y) are in A159589.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159589, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

Programs

  • Magma
    I:=[0, 31, 1204, 1347, 1504, 8151, 8980]; [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, May 08 2018
  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 31, 1204, 1347, 1504, 8151, 8980}, 50] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
  • PARI
    {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+898*n+201601), print1(n, ",")))}
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(31+1173*x+143*x^2-29*x^3-391*x^4 -29*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, May 08 2018

Formula

a(n) = 6*a(n-3) -a(n-6) +898 for n > 6; a(1)=0, a(2)=31, a(3)=1204, a(4)=1347, a(5)=1504, a(6)=8151.
G.f.: x*(31+1173*x+143*x^2-29*x^3-391*x^4-29*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 449*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 17 2009

A159589 Positive numbers y such that y^2 is of the form x^2+(x+449)^2 with integer x.

Original entry on oeis.org

421, 449, 481, 2045, 2245, 2465, 11849, 13021, 14309, 69049, 75881, 83389, 402445, 442265, 486025, 2345621, 2577709, 2832761, 13671281, 15023989, 16510541, 79682065, 87566225, 96230485, 464421109, 510373361, 560872369, 2706844589
Offset: 1

Views

Author

Klaus Brockhaus, Apr 18 2009

Keywords

Comments

(-29,a(1)) and (A130004(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+449)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (451+30*sqrt(2))/449 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (507363+329222*sqrt(2))/449^2 for n mod 3 = 1.

Examples

			(-29, a(1)) = (-29, 421) is a solution: (-29)^2+(-29+449)^2 = 841+176400 = 177241 = 421^2.
(A130004(1), a(2)) = (0, 449) is a solution: 0^2+(0+449)^2 = 201601 = 449^2.
(A130004(3), a(4)) = (1204, 2045) is a solution: 1204^2+(1204+449)^2 = 1449616+2732409 = 4182025 = 2045^2.

Links

Crossrefs

Cf. A130004, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159590 (decimal expansion of (451+30*sqrt(2))/449), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

Programs

  • Magma
    I:=[421,449,481,2045,2245,2465]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, May 08 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1}, {421,449,481,2045,2245,2465}, 50] (* G. C. Greubel, May 08 2018 *)
  • PARI
    {forstep(n=-32, 50000000, [3, 1], if(issquare(2*n^2+898*n+201601, &k), print1(k, ",")))}
  • PARI
    x='x+O('x^30); Vec((1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4)/(1- 6*x^3+x^6)) \\ G. C. Greubel, May 08 2018

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=421, a(2)=449, a(3)=481, a(4)=2045, a(5)=2245, a(6)=2465.
G.f.: (1-x)*(421+870*x+1351*x^2+870*x^3+421*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 449*A001653(k) for k >= 1.

A159590 Decimal expansion of (451+30*sqrt(2))/449.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 17 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A130004.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A159589.

Examples

			(451+30*sqrt(2))/449 = 1.09894522688461659568...

Links

Crossrefs

Cf. A130004, A159589, A002193 (decimal expansion of sqrt(2)), A159591 (decimal expansion of (507363+329222*sqrt(2))/449^2).

Programs

  • Magma
    (451+30*Sqrt(2))/449; // G. C. Greubel, May 05 2018
  • Mathematica
    RealDigits[(451+30*Sqrt[2])/449, 10, 100][[1]] (* G. C. Greubel, May 05 2018 *)
  • PARI
    (451+30*sqrt(2))/449 \\ G. C. Greubel, May 05 2018

Formula

Equals (30+sqrt(2))/(30-sqrt(2)).
Showing 1-3 of 3 results.

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