A157120 -id:A157120 - cp's OEIS frontend

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

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

Klaus Brockhaus, Feb 25 2009

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.

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

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

11-3*Sqrt(2) // G. C. Greubel, Jan 27 2018

RealDigits[11 - 3*Sqrt[2], 10, 50][[1]] (* G. C. Greubel, Jan 27 2018 *)

11-sqrt(18) \\ Charles R Greathouse IV, May 07 2015

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

Klaus Brockhaus, Feb 25 2009

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.

Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

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))).

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

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.

A157121 Decimal expansion of 11+3*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

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.

			11+3*sqrt(2) = 15.24264068711928514640...

Muniru A Asiru, Table of n, a(n) for n = 2..2000
Index entries for algebraic numbers, degree 2.

Cf. A157119, A157120, A157122 (decimal expansion of 11-3*sqrt(2)), A157123 (decimal expansion of (11+3*sqrt(2))/(11-3*sqrt(2))).

evalf[120](11+3*sqrt(2)); # Muniru A Asiru, Feb 12 2019

RealDigits[11+3*Sqrt[2],10,120][[1]] (* Harvey P. Dale, Sep 12 2012 *)

Equals 7+A083729 = 11+A010474. [R. J. Mathar, Feb 27 2009]

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

Klaus Brockhaus, Feb 25 2009

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.

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

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for algebraic numbers, degree 2.

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

(11+3*Sqrt(2))/(11-3*Sqrt(2)) // G. C. Greubel, Jan 27 2018

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

(11+3*sqrt(2))/(11-3*sqrt(2)) \\ G. C. Greubel, Jan 27 2018

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

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

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A157121 Decimal expansion of 11+3*sqrt(2).

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

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