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.

A129298 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.

Original entry on oeis.org

0, 51, 120, 267, 540, 931, 1780, 3367, 5644, 10591, 19840, 33111, 61944, 115851, 193200, 361251, 675444, 1126267, 2105740, 3936991, 6564580, 12273367, 22946680, 38261391, 71534640, 133743267, 223003944, 416934651, 779513100, 1299762451
Offset: 1

Views

Author

Mohamed Bouhamida, May 26 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+89, y).
Corresponding values y of solutions (x, y) are in A160055.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A160055, A129289, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

Programs

  • Magma
    I:=[0,51,120,267,540,931,1780]; [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 19 2018
  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,51,120,267,540,931,1780},30] (* Harvey P. Dale, Sep 21 2013 *)
  • PARI
    {forstep(n=0, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921), print1(n, ",")))};
  • PARI
    x='x+O('x^30); concat(0, Vec(x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5)/((1-x)*(1-6*x^3+x^6)))) \\ G. C. Greubel, Apr 19 2018

Formula

a(n) = 6*a(n-3) - a(n-6) + 178 with for n > 6; a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931.
G.f.: x*(51+69*x+147*x^2-33*x^3-23*x^4-33*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 89*A001652(k), k >= 0. (Zak Seidov, May 28 2007)
a(1)=0, a(2)=51, a(3)=120, a(4)=267, a(5)=540, a(6)=931, a(7)=1780, a(n) = a(n-1) + 6*a(n-3) - 6*a(n-4) - a(n-6) + a(n-7). - Harvey P. Dale, Sep 21 2013

Extensions

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

A160055 Positive numbers y such that y^2 is of the form x^2+(x+89)^2 with integer x.

Original entry on oeis.org

65, 89, 149, 241, 445, 829, 1381, 2581, 4825, 8045, 15041, 28121, 46889, 87665, 163901, 273289, 510949, 955285, 1592845, 2978029, 5567809, 9283781, 17357225, 32451569, 54109841, 101165321, 189141605, 315375265, 589634701, 1102398061
Offset: 1

Views

Author

Klaus Brockhaus, May 04 2009

Keywords

Comments

(-33, a(1)) and (A129298(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+89)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (107+42*sqrt(2))/89 for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (8979+2990*sqrt(2))/89^2 for n mod 3 = 1.

Examples

			(-33, a(1)) = (-33, 65) is a solution: (-33)^2+(-33+89)^2 = 1089+3136 = 4225 = 65^2.
(A129298(1), a(2)) = (0, 89) is a solution: 0^2+(0+89)^2 = 7921 = 89^2.
(A129298(3), a(4)) = (120, 241) is a solution: 120^2+(120+89)^2 = 14400+43681 = 58081 = 241^2.

Links

Crossrefs

Cf. A129298, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A160056 (decimal expansion of (107+42*sqrt(2))/89), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

Programs

  • Mathematica
    LinearRecurrence[{0,0,6,0,0,-1},{65,89,149,241,445,829},40] (* Harvey P. Dale, Feb 04 2015 *)
  • PARI
    {forstep(n=-36, 10000000, [3, 1], if(issquare(2*n^2+178*n+7921, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=65, a(2)=89, a(3)=149, a(4)=241, a(5)=445, a(6)=829.
G.f.: (1-x)*(65+154*x+303*x^2+154*x^3+65*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 89*A001653(k) for k >= 1.

A160056 Decimal expansion of (107+42*sqrt(2))/89.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 04 2009

Keywords

Comments

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129298.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A160055.

Examples

			(107+42*sqrt(2))/89 = 1.86962887213112350617...

Links

Crossrefs

Cf. A129298, A160055, A002193 (decimal expansion of sqrt(2)), A160057 (decimal expansion of (8979+2990*sqrt(2))/89^2).

Programs

  • Magma
    (107+42*Sqrt(2))/89; // G. C. Greubel, Apr 15 2018
  • Mathematica
    RealDigits[(107+42*Sqrt[2])/89, 10, 100][[1]] (* G. C. Greubel, Apr 15 2018 *)
  • PARI
    (107+42*sqrt(2))/89 \\ G. C. Greubel, Apr 15 2018

Formula

Equals (14+3*sqrt(2))/(14-3*sqrt(2)).
Showing 1-3 of 3 results.

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