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 1-4 of 4 results.

A157122 Decimal expansion of 11 - 3*sqrt(2).

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Feb 25 2009

Keywords

Comments

lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}, b = A157119.
lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}, b = A157120.

Examples

			11 - 3*sqrt(2) = 6.75735931288071485359...

Links

Crossrefs

Cf. A157119, A157120, A157121 (decimal expansion of 11+3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Programs

A157119 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

Original entry on oeis.org

0, 84, 105, 309, 765, 884, 2060, 4712, 5405, 12257, 27713, 31752, 71688, 161772, 185313, 418077, 943125, 1080332, 2436980, 5497184, 6296885, 14204009, 32040185, 36701184, 82787280, 186744132, 213910425, 482519877, 1088424813, 1246761572
Offset: 1

Views

Author

Klaus Brockhaus, Feb 25 2009

Keywords

Comments

Corresponding values y of solutions (x, y) are in A157120.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 0.

Links

Crossrefs

Cf. A157120, A001652, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Programs

  • PARI
    {forstep(n=0, 1300000000, [1, 3], if(issquare(2*n^2+206*n+10609), print1(n, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6)+206 for n > 6; a(1) = 0, a(2) = 84, a(3) = 105, a(4) = 309, a(5) = 765, a(6) = 884.
G.f.: x*(84+21*x+204*x^2-48*x^3-7*x^4-48*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 103*A001652(k) for k >= 0.

A157123 Decimal expansion of (11 + 3*sqrt(2))/(11 - 3*sqrt(2)).

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Feb 25 2009

Keywords

Comments

Lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {1, 2}, b = A157119.
Lim_{n -> infinity} b(n)/b(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}, b = A157120.

Examples

			(11 + 3*sqrt(2))/(11 - 3*sqrt(2)) = 2.25570966132644925457...

Links

Crossrefs

Cf. A157119, A157120, A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)).

Programs

  • Magma
    (11+3*Sqrt(2))/(11-3*Sqrt(2)) // G. C. Greubel, Jan 27 2018
  • Mathematica
    RealDigits[(11 + 3*Sqrt[2])/(11 - 3*Sqrt[2]), 10, 100][[1]] (* G. C. Greubel, Jan 27 2018 *)
  • PARI
    (11+3*sqrt(2))/(11-3*sqrt(2)) \\ G. C. Greubel, Jan 27 2018

A157120 Positive numbers y such that y^2 is of the form x^2+(x+103)^2 with integer x.

Original entry on oeis.org

73, 103, 205, 233, 515, 1157, 1325, 2987, 6737, 7717, 17407, 39265, 44977, 101455, 228853, 262145, 591323, 1333853, 1527893, 3446483, 7774265, 8905213, 20087575, 45311737, 51903385, 117078967, 264096157, 302515097, 682386227, 1539265205
Offset: 1

Views

Author

Klaus Brockhaus, Feb 25 2009

Keywords

Comments

(-48, a(1)) and (A157119(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+103)^2 = y^2.

Examples

			(-48, a(1)) = (-48, 73) is a solution: (-48)^2+(-48+103)^2 = 2304+3025 = 5329 = 73^2.
(A157119(1), a(2)) = (0, 103) is a solution: 0^2+(0+103)^2 = 10609 = 103^2,
(A157119(3), a(4)) = (105, 233) is a solution: 105^2+(105+103)^2 = 11025+43264 = 54289 = 233^2.

Links

Crossrefs

Cf. A157119, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157121 (decimal expansion of 11+3*sqrt(2)), A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

Programs

  • Mathematica
    Select[Table[Sqrt[x^2+(x+103)^2],{x,-50,3*10^6}],IntegerQ] (* THe program generates the first 20 terms of the sequence. *) (* or *) LinearRecurrence[ {0,0,6,0,0,-1},{73,103,205,233,515,1157},50](* Harvey P. Dale, Aug 19 2020 *)
  • PARI
    {forstep(n=-48, 1100000000, [1, 3], if(issquare(2*n^2+206*n+10609, &k), print1(k, ",")))}

Formula

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1) = 73, a(2) = 103, a(3) = 205, a(4) = 233, a(5) = 515, a(6) = 1157.
G.f.: x*(1-x)*(73+176*x+381*x^2+176*x^3+73*x^4) / (1-6*x^3+x^6).
a(3*k-1) = 103*A001653(k) for k >= 1.
Limit_{n -> oo} a(n)/a(n-3) = 3+2*sqrt(2).
Limit_{n -> oo} a(n)/a(n-1) = (3+2*sqrt(2))*(11-3*sqrt(2))^2/(11+3*sqrt(2))^2 for n mod 3 = 1.
Limit_{n -> oo} a(n)/a(n-1) = (11+3*sqrt(2))/(11-3*sqrt(2)) for n mod 3 = {0, 2}.

Extensions

Typo corrected by Klaus Brockhaus, Mar 01 2009
Showing 1-4 of 4 results.

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