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.

A101152 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.

Original entry on oeis.org

0, 111, 1260, 1707, 2280, 8791, 11380, 14707, 52624, 67711, 87100, 308091, 396024, 509031, 1797060, 2309571, 2968224, 10475407, 13462540, 17301451, 61056520, 78466807, 100841620, 355864851, 457339440, 587749407, 2074133724, 2665570971
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+569, y).
Corresponding values y of solutions (x, y) are in A160090.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160090, A129298, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).

Programs

  • Magma
    I:=[0,111,1260,1707,2280,8791,11380]; [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, Apr 21 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,111,1260,1707,2280,8791,11380}, 50] (* G. C. Greubel, Apr 21 2018 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761), print1(n, ",")))}
  • PARI
    x='x+O('x^30); concat([0], Vec(x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)))) \\ G. C. Greubel, Apr 21 2018

Formula

a(n) = 6*a(n-3) - a(n-6) + 1138 for n > 6; a(1)=0, a(2)=111, a(3)=1260, a(4)=1707, a(5)=2280, a(6)=8791.
G.f.: x*(111 +1149*x +447*x^2 -93*x^3 -383*x^4 -93*x^5)/((1-x)*(1-6*x^3 +x^6)).
a(3*k+1) = 569*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 04 2009

A160090 Positive numbers y such that y^2 is of the form x^2 + (x + 569)^2 with integer x.

Original entry on oeis.org

485, 569, 689, 2221, 2845, 3649, 12841, 16501, 21205, 74825, 96161, 123581, 436109, 560465, 720281, 2541829, 3266629, 4198105, 14814865, 19039309, 24468349, 86347361, 110969225, 142611989, 503269301, 646776041, 831203585, 2933268445
Offset: 1

Views

Author

Klaus Brockhaus, May 04 2009

Keywords

Comments

(-93, a(1)) and (A101152(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+569)^2 = y^2.
Lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
Lim_{n -> infinity} a(n)/a(n-1) = (587+102*sqrt(2))/569 for n mod 3 = {0, 2}.
Lim_{n -> infinity} a(n)/a(n-1) = (617139+371510*sqrt(2))/569^2 for n mod 3 = 1.

Examples

			(-93, a(1)) = (-93, 485) is a solution: (-93)^2+(-93+569)^2 = 8649+226576 = 235225 = 485^2.
(A101152(1), a(2)) = (0, 569) is a solution: 0^2+(0+569)^2 = 323761= 569^2.
(A101152(3), a(4)) = (1260, 2221) is a solution: 1260^2+(1260+569)^2 = 1587600+3345241 = 4932841 = 2221^2.

Links

Crossrefs

Cf. A101152, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569), A160092 (decimal expansion of (617139+371510*sqrt(2))/569^2).

Programs

  • Magma
    I:=[485,569,689,2221,2845,3649]; [n le 6 select I[n] else 6*Self(n-3) - Self(n-6): n in [1..30]]; // G. C. Greubel, Apr 21 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1}, {485,569,689,2221,2845,3649}, 50] (* G. C. Greubel, Apr 21 2018 *)
  • PARI
    {forstep(n=-96, 10000000, [3, 1], if(issquare(2*n^2+1138*n+323761, &k), print1(k, ",")))}
  • PARI
    x='x+O('x^30); Vec((1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, Apr 21 2018

Formula

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=485, a(2)=569, a(3)=689, a(4)=2221, a(5)=2845, a(6)=3649.
G.f.: (1-x)*(485 +1054*x +1743*x^2 +1054*x^3 +485*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 569*A001653(k) for k >= 1.

A160092 Decimal expansion of (617139 + 371510*sqrt(2))/569^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 04 2009

Keywords

Comments

Equals Lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A101152.
Equals Lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160090.

Examples

			(617139+371510*sqrt(2))/569^2 = 3.52894104156222812994...

Links

Crossrefs

Cf. A101152, A160090, A002193 (decimal expansion of sqrt(2)), A160091 (decimal expansion of (587+102*sqrt(2))/569).

Programs

  • Magma
    (617139 +371510*Sqrt(2))/569^2; // G. C. Greubel, Apr 21 2018
  • Mathematica
    RealDigits[(617139 +371510*Sqrt[2])/569^2, 10, 100][[1]] (* G. C. Greubel, Apr 21 2018 *)
  • PARI
    (617139 +371510*sqrt(2))/569^2 \\ G. C. Greubel, Apr 21 2018

Formula

Equals (1940 + 766*sqrt(2))/(1940 - 766*sqrt(2)).
Equals (3 + 2*sqrt(2))*(34 - 3*sqrt(2))^2/(34 + 3*sqrt(2))^2.
Showing 1-3 of 3 results.

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