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.

A115135 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.

Original entry on oeis.org

0, 108, 1407, 1851, 2407, 9768, 12340, 15568, 58435, 73423, 92235, 342076, 429432, 539076, 1995255, 2504403, 3143455, 11630688, 14598220, 18322888, 67790107, 85086151, 106795107, 395111188, 495919920, 622448988, 2302878255, 2890434603
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 03 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+617, y).
Corresponding values y of solutions (x, y) are in A160176.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160176, A001652, A111258, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).

Programs

  • Magma
    I:=[0,108,1407,1851,2407,9768,12340]; [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 04 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,108,1407,1851,2407,9768,12340}, 50] (* G. C. Greubel, May 04 2018 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689), print1(n, ",")))}
  • PARI
    x='x+O('x^30); Vec(x*(108 +1299*x +444*x^2 -92*x^3 -433*x^4 -92*x^5)/((1-x)*(1 -6*x^3 +x^6))) \\ G. C. Greubel, May 04 2018

Formula

a(n) = 6*a(n-3) -a(n-6) +1234 for n > 6; a(1)=0, a(2)=108, a(3)=1407, a(4)=1851, a(5)=2407, a(6)=9768.
G.f.: x*(108 +1299*x +444*x^2 -92*x^3 -433*x^4 -92*x^5)/((1-x)*(1 -6*x^3 +x^6)).
a(3*k+1) = 617*A001652(k) for k >= 0.

Extensions

Edited and two terms added by Klaus Brockhaus, May 18 2009

A160176 Positive numbers y such that y^2 is of the form x^2+(x+617)^2 with integer x.

Original entry on oeis.org

533, 617, 733, 2465, 3085, 3865, 14257, 17893, 22457, 83077, 104273, 130877, 484205, 607745, 762805, 2822153, 3542197, 4445953, 16448713, 20645437, 25912913, 95870125, 120330425, 151031525, 558772037, 701337113, 880276237
Offset: 1

Views

Author

Klaus Brockhaus, May 18 2009

Keywords

Comments

(-92, a(1)) and (A115135(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+617)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (633+100*sqrt(2))/617 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (755667+461578*sqrt(2))/617^2 for n mod 3 = 1.

Examples

			(-92, a(1)) = (-92, 533) is a solution: (-92)^2+(-92+617)^2 = 8464+275625 = 284089 = 533^2.
(A115135(1), a(2)) = (0, 617) is a solution: 0^2+(0+617)^2 = 380689 = 617^2.
(A115135(3), a(4)) = (1407, 2465) is a solution: 1407^2+(1407+617)^2 = 1979649+4096576 = 6076225 = 2465^2.

Links

Crossrefs

Cf. A115135, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617), A160178 (decimal expansion of (755667+461578*sqrt(2))/617^2).

Programs

  • Magma
    I:=[533,617,733,2465,3085,3865]; [n le 6 select I[n] else 6*Self(n31) -Self(n-6): n in [1..30]]; // G. C. Greubel, May 04 2018
  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1}, {533,617,733,2465,3085,3865}, 50] (* G. C. Greubel, May 04 2018 *)
  • PARI
    {forstep(n=-92, 10000000, [3, 1], if(issquare(2*n^2+1234*n+380689, &k), print1(k, ",")))}
  • PARI
    x='x+O('x^30); Vec((1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6)) \\ G. C. Greubel, May 04 2018

Formula

a(n) = 6*a(n-3) -a(n-6) for n > 6; a(1)=533, a(2)=617, a(3)=733, a(4)=2465, a(5)=3085, a(6)=3865.
G.f.: (1-x)*(533 +1150*x +1883*x^2 +1150*x^3 +533*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 617*A001653(k) for k >= 1.

A160178 Decimal expansion of (755667+461578*sqrt(2))/617^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 18 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 0, b = A115135.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = 1, b = A160176.

Examples

			(755667+461578*sqrt(2))/617^2 = 3.69970466100425404053...

Links

Crossrefs

Cf. A115135, A160176, A002193 (decimal expansion of sqrt(2)), A160177 (decimal expansion of (633+100*sqrt(2))/617).

Programs

  • Magma
    (755667+461578*Sqrt(2))/617^2; // G. C. Greubel, Apr 25 2018
  • Mathematica
    RealDigits[(755667+461578*Sqrt[2])/617^2,10,120][[1]] (* Harvey P. Dale, Oct 14 2017 *)
  • PARI
    (755667+461578*sqrt(2))/617^2 \\ G. C. Greubel, Apr 25 2018

Formula

Equals (1066 +433*sqrt(2))/(1066 -433*sqrt(2)).
Equals (3 +2*sqrt(2))*(25 -2*sqrt(2))^2/(25 +2*sqrt(2))^2.
Showing 1-3 of 3 results.

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