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.

A130017 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.

Original entry on oeis.org

0, 45, 2688, 2901, 3128, 18105, 19340, 20657, 107876, 115073, 122748, 631085, 673032, 717765, 3680568, 3925053, 4185776, 21454257, 22879220, 24398825, 125046908, 133352201, 142209108, 728829125, 777235920, 828857757, 4247929776
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+967, y).
Corresponding values y of solutions (x, y) are in A159701.
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) = (969+44**sqrt(2))/967 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159701, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44**sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,45,2688,2901,3128,18105,19340},40] (* Harvey P. Dale, Nov 03 2013 *)
  • PARI
    {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1934 for n > 6; a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105.
G.f.: x*(45+2643*x+213*x^2-43*x^3-881*x^4-43*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 967*A001652(k) for k >= 0.
a(1)=0, a(2)=45, a(3)=2688, a(4)=2901, a(5)=3128, a(6)=18105, a(7)=19340, a(n)=a(n-1)+6*a(n-3)-6*a(n-4)-a(n-6)+a(n-7). - Harvey P. Dale, Nov 03 2013

Extensions

Edited and two terms added by Klaus Brockhaus, Apr 21 2009

A159701 Positive numbers y such that y^2 is of the form x^2+(x+967)^2 with integer x.

Original entry on oeis.org

925, 967, 1013, 4537, 4835, 5153, 26297, 28043, 29905, 153245, 163423, 174277, 893173, 952495, 1015757, 5205793, 5551547, 5920265, 30341585, 32356787, 34505833, 176843717, 188589175, 201114733, 1030720717, 1099178263
Offset: 1

Views

Author

Klaus Brockhaus, Apr 21 2009

Keywords

Comments

(-43, a(1)) and (A130017(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+967)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (969+44*sqrt(2))/967 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (2487411+1629850*sqrt(2))/967^2 for n mod 3 = 1.

Examples

			(-43, a(1)) = (-43, 925) is a solution: (-43)^2+(-43+967)^2 = 1849+853776 = 855625 = 925^2.
(A130017(1), a(2)) = (0, 967) is a solution: 0^2+(0+967)^2 = 935089 = 967^2.
(A130017(3), a(4)) = (2688, 4537) is a solution: 2688^2+(2688+967)^2 = 7225344+13359025 = 20584369 = 4537^2.

Links

Crossrefs

Cf. A130017, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A159702 (decimal expansion of (969+44*sqrt(2))/967), A159703 (decimal expansion of (2487411+1629850*sqrt(2))/967^2).

Programs

  • Magma
    I:=[925,967,1013,4537,4835,5153]; [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}, {925,967,1013,4537,4835,5153}, 40] (* G. C. Greubel, May 22 2018 *)
  • PARI
    {forstep(n=-44, 10000000, [1, 3], if(issquare(2*n^2+1934*n+935089, &k), print1(k, ",")))};
  • PARI
    x='x+O('x^30); Vec((1-x)*(925+1892*x+2905*x^2+1892*x^3+925*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)=925, a(2)=967, a(3)=1013, a(4)=4537, a(5)=4835, a(6)=5153.
G.f.: (1-x)*(925+1892*x+2905*x^2+1892*x^3+925*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 967*A001653(k) for k >= 1.

A159703 Decimal expansion of (2487411+1629850*sqrt(2))/967^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 21 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A130017.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A159701.

Examples

			(2487411+1629850*sqrt(2))/967^2 = 5.12503833820501467270...

Links

Crossrefs

Cf. A130017, A159701, A002193 (decimal expansion of sqrt(2)), A159702 (decimal expansion of (969+44*sqrt(2))/967).

Programs

  • Magma
    (2487411 +1629850*Sqrt(2))/967^2; // G. C. Greubel, May 22 2018
  • Mathematica
    RealDigits[(2487411+1629850*Sqrt[2])/967^2, 10, 100][[1]] (* G. C. Greubel, May 22 2018 *)
  • PARI
    (2487411 +1629850*sqrt(2))/967^2 \\ G. c. Greubel, May 22 2018

Formula

Equals (1850 +881*sqrt(2))/(1850 -881*sqrt(2)).
Equals (3 +2*sqrt(2))*(44 -sqrt(2))^2/(44 +sqrt(2))^2.
Showing 1-3 of 3 results.

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