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.

A129991 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

Original entry on oeis.org

0, 23, 620, 723, 840, 4223, 4820, 5499, 25200, 28679, 32636, 147459, 167736, 190799, 860036, 978219, 1112640, 5013239, 5702060, 6485523, 29219880, 33234623, 37800980, 170306523, 193706160, 220320839, 992619740, 1129002819, 1284124536, 5785412399, 6580311236
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 14 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+241, y).
Corresponding values y of solutions (x, y) are in A159565.
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) = (243+22*sqrt(2))/241 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159565, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

Programs

  • Mathematica
    LinearRecurrence[{1, 0, 6, -6, 0, -1, 1}, {0, 23, 620, 723, 840, 4223, 4820}, 40] (* Vladimir Joseph Stephan Orlovsky, Feb 14 2012 *)
  • PARI
    {forstep(n=0, 500000000, [3, 1], if(issquare(2*n^2+482*n+58081), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+482 for n > 6; a(1)=0, a(2)=23, a(3)=620, a(4)=723, a(5)=840, a(6)=4223.
G.f.: x*(23+597*x+103*x^2-21*x^3-199*x^4-21*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 241*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 16 2009

A159567 Decimal expansion of (137283+87958*sqrt(2))/241^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 16 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A129991.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159565.

Examples

			(137283+87958*sqrt(2))/241^2 = 4.50533559200448846098...

Links

Crossrefs

Cf. A129991, A159565, A002193 (decimal expansion of sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241).

Programs

  • Magma
    (137283+87958*Sqrt(2))/241^2; // G. C. Greubel, May 05 2018
  • Mathematica
    RealDigits[(137283+87958*Sqrt[2])/241^2, 10, 100][[1]] (* G. C. Greubel, May 05 2018 *)
  • PARI
    (137283+87958*sqrt(2))/241^2 \\ G. C. Greubel, May 05 2018

Formula

Equals (442+199*sqrt(2))/(442-199*sqrt(2)).
Equals (3+2*sqrt(2))*(22-sqrt(2))^2/(22+sqrt(2))^2.

A159565 Positive numbers y such that y^2 is of the form x^2+(x+241)^2 with integer x.

Original entry on oeis.org

221, 241, 265, 1061, 1205, 1369, 6145, 6989, 7949, 35809, 40729, 46325, 208709, 237385, 270001, 1216445, 1383581, 1573681, 7089961, 8064101, 9172085, 41323321, 47001025, 53458829, 240849965, 273942049, 311580889, 1403776469, 1596651269
Offset: 1

Views

Author

Klaus Brockhaus, Apr 16 2009

Keywords

Comments

(-21,a(1)) and (A129991(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+241)^2 = y^2.

Examples

			(-21, a(1)) = (-21, 221) is a solution: (-21)^2+(-21+241)^2 = 441+48400 = 48841 = 221^2.
(A129993(1), a(2)) = (0, 241) is a solution: 0^2+(0+241)^2 = 58081= 241^2.
(A129993(3), a(4)) = (620, 1061) is a solution: 620^2+(620+241)^2 = 384400+741321 = 1125721 = 1061^2.

Links

Crossrefs

Cf. A129991, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159566 (decimal expansion of (243+22*sqrt(2))/241), A159567 (decimal expansion of (137283+87958*sqrt(2))/241^2).

Programs

  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1},{221,241,265,1061,1205,1369},30] (* Harvey P. Dale, Nov 21 2011 *)
  • PARI
    {forstep(n=-24, 50000000, [3, 1], if(issquare(2*n^2+482*n+58081, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=221, a(2)=241, a(3)=265, a(4)=1061, a(5)=1205, a(6)=1369.
G.f.: x*(1-x)*(221+462*x+727*x^2+462*x^3+221*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 241*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (243+22*sqrt(2))/241 for n mod 3 = {0, 2}.
Limit_{n -> oo} a(n)/a(n-1) = (137283+87958*sqrt(2))/241^2 for n mod 3 = 1.
Showing 1-3 of 3 results.

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