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.

A130014 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+881)^2 = y^2.

Original entry on oeis.org

0, 43, 2440, 2643, 2860, 16443, 17620, 18879, 97980, 104839, 112176, 573199, 613176, 655939, 3342976, 3575979, 3825220, 19486419, 20844460, 22297143, 113577300, 121492543, 129959400, 661979143, 708112560, 757461019, 3858299320
Offset: 1

Views

Author

Mohamed Bouhamida, Jun 15 2007

Keywords

Comments

Also values x of Pythagorean triples (x, x+881, y).
Corresponding values y of solutions (x, y) are in A159690.
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) = (883+42*sqrt(2))/881 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A159690, A066436, A118673, A118674, A129836, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

Programs

  • Mathematica
    LinearRecurrence[{1,0,6,-6,0,-1,1},{0,43,2440,2643,2860,16443,17620},30] (* Harvey P. Dale, Aug 13 2015 *)
  • PARI
    {forstep(n=0, 10000000, [1, 3], if(issquare(2*n^2+1762*n+776161), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+1762 for n > 6; a(1)=0, a(2)=43, a(3)=2440, a(4)=2643, a(5)=2860, a(6)=16443.
G.f.: x*(43+2397*x+203*x^2-41*x^3-799*x^4-41*x^5) / ((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 881*A001652(k) for k >= 0.

Extensions

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

A159691 Decimal expansion of (883 + 42*sqrt(2))/881.

Original entry on oeis.org

1, 0, 6, 9, 6, 9, 0, 0, 9, 0, 3, 7, 4, 1, 9, 9, 7, 6, 3, 9, 6, 1, 0, 3, 3, 9, 6, 8, 6, 9, 1, 0, 4, 1, 2, 2, 5, 0, 8, 5, 0, 4, 6, 7, 8, 6, 3, 4, 0, 3, 3, 1, 2, 3, 6, 1, 7, 8, 6, 8, 3, 9, 3, 7, 5, 6, 5, 9, 6, 0, 0, 0, 4, 6, 4, 8, 5, 9, 0, 8, 0, 0, 9, 4, 4, 0, 5, 9, 3, 3, 7, 6, 1, 8, 3, 4, 3, 8, 6, 6, 1, 2, 3, 5, 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 = {1, 2}, b = A130014.
Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A159690.

Examples

			(883 + 42*sqrt(2))/881 = 1.06969009037419976396...

Links

Crossrefs

Cf. A130014, A159690, A002193 (decimal expansion of sqrt(2)), A159692 (decimal expansion of (2052963+1343918*sqrt(2))/881^2).

Programs

  • Magma
    (883+42*Sqrt(2))/881; // G. C. Greubel, May 10 2018
  • Mathematica
    RealDigits[(883+42*Sqrt[2])/881, 10, 100][[1]] (* G. C. Greubel, May 10 2018 *)
  • PARI
    (883+42*sqrt(2))/881 \\ G. C. Greubel, May 10 2018

Formula

Equals (42 + sqrt(2))/(42 - sqrt(2)).

A159692 Decimal expansion of (2052963 + 1343918*sqrt(2))/881^2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 21 2009

Keywords

Comments

Lim_{n -> infinity} b(n)/b(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 0, b = A130014.
Lim_{n -> infinity} b(n)/b(n-1) = (2052963+1343918*sqrt(2))/881^2 for n mod 3 = 1, b = A159690.

Examples

			(2052963 + 1343918*sqrt(2))/881^2 = 5.09372419165266633056...

Links

Crossrefs

Cf. A130014, A159690, A002193 (decimal expansion of sqrt(2)), A159691 (decimal expansion of (883+42*sqrt(2))/881).

Programs

  • Magma
    (2052963 + 1343918*Sqrt(2))/881^2; // G. C. Greubel, Jun 02 2018
  • Mathematica
    RealDigits[(2052963 + 1343918*Sqrt[2])/881^2, 10, 100][[1]] (* G. C. Greubel, Jun 02 2018 *)
  • PARI
    (2052963 + 1343918*sqrt(2))/881^2 \\ G. C. Greubel, Jun 02 2018

Formula

Equals (1682 + 799*sqrt(2))/(1682 - 799*sqrt(2)).
Equals (3 + 2*sqrt(2))*(42 - sqrt(2))^2/(42 + sqrt(2))^2.
Showing 1-3 of 3 results.

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