A157215 -id:A157215 - cp's OEIS frontend

A268682 Decimal expansion of 1 - 1/sqrt(2).

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

Offset: 0

Charles R Greathouse IV, Feb 19 2016

This is the maximum fraction of mass-energy of a black hole which can come from angular momentum, and hence the maximum energy which can be extracted from the black hole via the Penrose process.
Differs from A157215 only in one or two leading digits. - R. J. Mathar, Feb 24 2016
This is the probability that a randomly selected vertex in a random Schroeder tree is a leaf as the number of leaves goes to infinity. See Corollary 2.1.2. of Van Duzer. - Michel Marcus, Apr 12 2019

			0.29289321881345247559915563789515096071516406231152596341166013100463376...

Charles D. Dermer and Govind Menon, High Energy Radiation from Black Holes: Gamma Rays, Cosmic Rays, and Neutrinos (2009). See pp. 400-402.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Anthony Van Duzer, Subtrees of a Given size in Schroeder Trees, arXiv:1904.05525 [math.CO], 2019.
Stijn J. van Tongeren, Rotating black holes (2009).
Wikipedia, Penrose process
Index entries for algebraic numbers, degree 2

Cf. A010503, A268683.

1-1/Sqrt(2); // Vincenzo Librandi, Feb 20 2016

RealDigits[1 - 1 / Sqrt[2], 10, 90] [[1]] (* Vincenzo Librandi, Feb 20 2016 *)

PARI
```
1-sqrt(.5)
```

Equals 1 - A010503.
a(n) = 9 - A010503(n). - Philippe Deléham, Feb 21 2016
Equals Integral_{x=0..Pi/4} sin(x) dx. - Amiram Eldar, Jun 29 2020

More digits from Jon E. Schoenfield, Mar 15 2018

A157214 Decimal expansion of 18 + 5*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}, b = A129544.

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}, b = A157213.

			18+5*sqrt(2) = 25.07106781186547524400...

G. C. Greubel, Table of n, a(n) for n = 2..5000
Index entries for algebraic numbers, degree 2

Cf. A129544, A157213, A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18-5*sqrt(2))/(18+5*sqrt(2))).

[18 + 5*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[18+5Sqrt[2],10,150][[1]]  (* Harvey P. Dale, Mar 18 2011 *)

18 + 5*sqrt(2) \\ G. C. Greubel, Nov 28 2017

Equals 18 + 10*A010503. - R. J. Mathar, Feb 27 2009

A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

Original entry on oeis.org

0, 115, 136, 411, 1036, 1155, 2740, 6375, 7068, 16303, 37488, 41527, 95352, 218827, 242368, 556083, 1275748, 1412955, 3241420, 7435935, 8235636, 18892711, 43340136, 48001135, 110115120, 252605155, 279771448, 641798283, 1472291068, 1630627827, 3740674852

Offset: 1

Mohamed Bouhamida, May 30 2007

Also values x of Pythagorean triples (x, x+137, y).
Corresponding values y of solutions (x, y) are in A157213.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(18-5*sqrt(2))^2/(18+5*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. A157213, A001652, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))), A129288, A129289, A129298.

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,115,136,411,1036,1155,2740},80] (* Vladimir Joseph Stephan Orlovsky, Feb 07 2012 *)

{forstep(n=0, 1500000000, [3, 1], if(issquare(2*n^2+274*n+18769), print1(n, ",")))}

a(n) = 6*a(n-3)-a(n-6)+274 for n > 6; a(1)=0, a(2)=115, a(3)=136, a(4)=411, a(5)=1036, a(6)=1155.
G.f.: x*(115+21*x+275*x^2-65*x^3-7*x^4-65*x^5)/((1-x)*(1-6*x^3+x^6)).
a(3*k+1) = 137*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 25 2009

A157216 Decimal expansion of (18 + 5sqrt(2))/(18 - 5sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 25 2009

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {1, 2}, b = A129544.

lim_{n -> infinity} b(n)/b(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}, b = A157213.

			(18 + 5*sqrt(2))/(18 - 5*sqrt(2)) = 2.29400890958816463059...

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

Cf. A129544, A157213, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)).

(18+5*Sqrt(2))/(18-5*Sqrt(2)) // G. C. Greubel, Jan 27 2018

With[{c=5Sqrt[2]},RealDigits[(18+c)/(18-c),10,120][[1]]] (* Harvey P. Dale, Dec 13 2011 *)

(18+5*sqrt(2))/(18-5*sqrt(2)) \\ G. C. Greubel, Jan 27 2018

A157213 Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x.

Original entry on oeis.org

97, 137, 277, 305, 685, 1565, 1733, 3973, 9113, 10093, 23153, 53113, 58825, 134945, 309565, 342857, 786517, 1804277, 1998317, 4584157, 10516097, 11647045, 26718425, 61292305, 67883953, 155726393, 357237733, 395656673, 907639933

Offset: 1

Klaus Brockhaus, Feb 25 2009

(-65, a(1)) and (A129544(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

			(-65, a(1)) = (-65, 97) is a solution: (-65)^2+(-65+137)^2 = 4225+5184 = 9409 = 97^2.
(A129544(1), a(2)) = (0, 137) is a solution: 0^2+(0+137)^2 = 18769 = 137^2.
(A129544(3), a(4)) = (136, 305) is a solution: 136^2+(136+137)^2 = 18496+74529 = 93025 = 305^2.

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

Cf. A129544, A001653, A157214 (decimal expansion of 18+5*sqrt(2)), A157215 (decimal expansion of 18-5*sqrt(2)), A157216 (decimal expansion of (18+5*sqrt(2))/(18-5*sqrt(2))).

{forstep(n=-68, 1000000000, [3, 1], if(issquare(n^2+(n+137)^2,&k), print1(k, ",")))}

a(n) = 6*a(n-3)-a(n-6) for n > 6; a(1)=97, a(2)=137, a(3)=277, a(4)=305, a(5)=685, a(6)=1565.
G.f.: x*(1-x)*(97+234*x+511*x^2+234*x^3+97*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 137*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))*(18-5*sqrt(2))^2/(18+5*sqrt(2))^2 for n mod 3 = 1.
Limit_{n -> oo} a(n)/a(n-1) = (18+5*sqrt(2))/(18-5*sqrt(2)) for n mod 3 = {0, 2}.

A335094 Decimal expansion of (15 - 4*sqrt(2))/8.

Original entry on oeis.org

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

Offset: 1

Jeremy Tan, Sep 12 2020

Largest overhang off an edge achievable with four unit-length bricks.

Corresponding values for one, two and three bricks are 1/2, 3/4 and 1 respectively.

			1.1678932188134524755991556378951...

S. Ainley, Finely Balanced, The Mathematical Gazette, 63 (1979), p. 272.
M. Paterson and U. Zwick, Overhang, arXiv:0710.2357 [math.HO], 2007.
Eric Weisstein's World of Mathematics, Book Stacking Problem

Cf. A020761 (1/2), A152627 (3/4).

Cf. A014537, A157215, A268682.

First@ RealDigits@ N[(15 - 4 Sqrt[2])/8, 102]

PARI
```
(15-4*sqrt(2))/8
```

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A268682 Decimal expansion of 1 - 1/sqrt(2).

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A157214 Decimal expansion of 18 + 5*sqrt(2).

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A129544 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+137)^2 = y^2.

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A157216 Decimal expansion of (18 + 5*sqrt(2))/(18 - 5*sqrt(2)).

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A157213 Positive numbers y such that y^2 is of the form x^2+(x+137)^2 with integer x.

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A335094 Decimal expansion of (15 - 4*sqrt(2))/8.

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A157216 Decimal expansion of (18 + 5sqrt(2))/(18 - 5sqrt(2)).