A157214 -id:A157214 - cp's OEIS frontend

A174968 Decimal expansion of (1 + sqrt(2))/2.

1, 2, 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: 1

Klaus Brockhaus, Apr 02 2010

a(n) is the diameter of the circle around the Vitruvian Man when the square has sides of unit length. See illustration in links. - Kival Ngaokrajang, Jan 29 2015
The iterated function z^2 - 1/4, starting from z = 0, gives a pretty good rational approximation of (-1)((1 + sqrt(2))/2 - 1) to more than eight decimal digits after just twenty steps. - Alonso del Arte, Apr 09 2016
This sequence describes the minimum Euclidean length of the optimal solution of the well-known Nine dots puzzle, published in Sam Loyd’s Cyclopedia of puzzles (1914), p. 301 since a valid polygonal chain satisfying the conditions of the above-mentioned problem is (0, 1)-(0, 3)-(3, 0)-(0, 0)-(2, 2), and its total length is equal to 5*(1 + sqrt(2)) = 12.071... (i.e., 10*(1 + sqrt(2))/2). - Marco Ripà, Jul 22 2024

			1.20710678118654752440084436210484903928483593768847...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Kival Ngaokrajang, Illustration of Vitruvian Man and construction rule.
Marco Ripà, Shortest polygonal chains covering each planar square grid, arXiv:2207.08708v3 [math.CO], 2024.
Wikipedia, Vitruvian Man.
Index entries for algebraic numbers, degree 2.

Cf. A002193 (decimal expansion of sqrt(2)), A010685 (continued fraction expansion of (1 + sqrt(2))/2), A079291, A249403.

Apart from initial digits the same as A157214 and A010503.

First@RealDigits[(1 + Sqrt[2])/2, 10, 105] (* Michael De Vlieger, Jan 29 2015 *)

(1+sqrt(2))/2 \\ Altug Alkan, Apr 16 2016

Equals Product_{k>=2} (1 + (-1)^k/A079291(k)). - Amiram Eldar, Dec 03 2024

A157215 Decimal expansion of 18 - 5*sqrt(2).

Original entry on oeis.org

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

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) = 10.92893218813452475599...

G. C. Greubel, Table of n, a(n) for n = 2..5001
Index entries for algebraic numbers, degree 2

Cf. A129544, A157213, A157214 (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 - 5*Sqrt[2], 10, 50][[1]] (* G. C. Greubel, Nov 28 2017 *)

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

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

A268683 Decimal expansion of (sqrt(2) - 1)/2.

Original entry on oeis.org

2, 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

Offset: 0

Charles R Greathouse IV, Feb 19 2016

This is the maximum increase in mass-energy a particle can carry away from a neutral rotating (Kerr) black hole via the Penrose process.

Apart from leading digits the same as A174968, A157214 and A010503. - R. J. Mathar, Feb 24 2016

			0.20710678118654752440084436210484903928483593768847403658833986899536623...

Subrahmanyan Chandrasekhar, The Mathematical Theory of Black Holes, Oxford (1983), pp. 368-369.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Stijn J. van Tongeren, Rotating black holes (2009).
Wikipedia, Penrose process
Index entries for algebraic numbers, degree 2.

Cf. A268682, A174968.

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

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

PARI
```
(sqrt(2)-1)/2
```

More digits from Jon E. Schoenfield, Mar 15 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}.

A351898 Decimal expansion of metallic ratio for N = 14.

Original entry on oeis.org

1, 4, 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

Offset: 2

A.H.M. Smeets, Feb 24 2022

Decimal expansion of continued fraction [14; 14, 14, 14, ...].
Also largest solution of x^2 - 14 x - 1 = 0.
Essentially the same digit sequence as A010503, A157214, A174968 and A268683.
The metallic ratio's for N = A077444(n) are equal to powers of the silver ratio, i.e., A014166^(2n-1); this constant represents the special case for N = A077444(2).

			14.0710678118654752440084436210484903928483593...

Michael Penn, 7+5√2 is a perfect cube??, YouTube video, 2022.
Wikipedia, Metallic mean.

Cf. A010503, A077444, A157214, A174968, A268683.

Metallic ratios: A001622 (N=1), A014176 (N=2), A098316 (N=3), A098317 (N=4), A098318 (N=5), A176398 (N=6), A176439 (N=7), A176458 (N=8), A176522 (N=9), A176537 (N=10), A244593 (N=11).

RealDigits[7 + 5*Sqrt[2], 10, 100][[1]] (* Amiram Eldar, Feb 24 2022 *)

PARI
```
(1+sqrt(2))^3
```

Equals 2 + 5*A014176.
Equals A014176^3.
Equals exp(arcsinh(7)). - Amiram Eldar, Jul 04 2023

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A174968 Decimal expansion of (1 + sqrt(2))/2.

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A157215 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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A268683 Decimal expansion of (sqrt(2) - 1)/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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A351898 Decimal expansion of metallic ratio for N = 14.

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