A157258 -id:A157258 - cp's OEIS frontend

A090488 Decimal expansion of 2 + 2*sqrt(2).

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

Offset: 1

Felix Tubiana, Feb 05 2004

Side length of smallest square containing five circles of radius 1. - Charles R Greathouse IV, Apr 05 2011
Equals n + n/(n +n/(n +n/(n +....))) for n = 4.  See also A090388. - Stanislav Sykora, Jan 23 2014
Also the area of a regular octagon with unit edge length. - Stanislav Sykora, Apr 12 2015
The positive solution to x^2 - 4*x - 4 = 0. The negative solution is -1 * A163960 = -0.82842... . - Michal Paulovic, Dec 12 2023

			4.828427124746190097603377448419396157139343750...

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

Cf. n+n/(n+n/(n+...)): A090388 (n=2), A090458 (n=3), A090550 (n=5), A092294 (n=6), A092290 (n=7), A090654 (n=8), A090655 (n=9), A090656 (n=10). - Stanislav Sykora, Jan 23 2014
Cf. A002193, A010466, A014176, A086178, A156035, A157258, A163960, A365823.
Cf. Areas of other regular polygons: A120011, A102771, A104956, A178817, A256853, A178816, A256854, A178809.

RealDigits[2+2Sqrt[2],10,120][[1]] (* Harvey P. Dale, Mar 11 2015 *)

2*(1 + sqrt(2)) \\ G. C. Greubel, Jul 03 2017

Equals 1 + A086178 = 2*A014176. - R. J. Mathar, Sep 03 2007
From Michal Paulovic, Dec 12 2023: (Start)
Equals A010466 + 2.
Equals A156035 - 1.
Equals A157258 - 5.
Equals A163960 + 4.
Equals A365823 - 2.
Equals [4; 1, 4, ...] (periodic continued fraction expansion).
Equals sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * sqrt(4 + 4 * ...)))). (End)

Better definition from Rick L. Shepherd, Jul 02 2004

A129288 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.

Original entry on oeis.org

0, 36, 39, 123, 319, 336, 820, 1960, 2059, 4879, 11523, 12100, 28536, 67260, 70623, 166419, 392119, 411720, 970060, 2285536, 2399779, 5654023, 13321179, 13987036, 32954160, 77641620, 81522519, 192071019, 452528623, 475148160, 1119472036

Offset: 1

Mohamed Bouhamida, May 26 2007

Also values x of Pythagorean triples (x, x+41, y).
Corresponding values y of solutions (x, y) are in A157257.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7 + 2*sqrt(2))/(7 - 2*sqrt(2)) for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3 + 2*sqrt(2))*(7 - 2*sqrt(2))^2/(7 + 2*sqrt(2))^2 for n mod 3 = 0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,6,-6,0,-1,1).

Cf. A157257, A001652, A156035 (decimal expansion of 3 + 2*sqrt(2)), A157258 (decimal expansion of 7 + 2*sqrt(2)), A157259 (decimal expansion of 7 - 2*sqrt(2)), A157260 (decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2))).

m:=25; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(36+3*x+84*x^2-20*x^3-x^4-20*x^5)/((1-x)*(1-6*x^3+ x^6)))); // G. C. Greubel, May 07 2018

LinearRecurrence[{1,0,6,-6,0,-1,1},{0,36,39,123,319,336,820},40] (* Harvey P. Dale, Jan 18 2015 *)

forstep(n=0, 1200000000, [3 ,1], if(issquare(2*n^2+82*n+1681), print1(n, ",")))

a(n) = 6*a(n-3) - a(n-6) + 82 for n > 6; a(1)=0, a(2)=36, a(3)=39, a(4)=123, a(5)=319, a(6)=336.
G.f.: x*(36 + 3*x + 84*x^2 - 20*x^3 - x^4 - 20*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 41*A001652(k) for k >= 0.

Edited and extended by Klaus Brockhaus, Feb 26 2009

A157259 Decimal expansion of 7 - 2*sqrt(2).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 26 2009

lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {1, 2}, b = A129288.

lim_{n -> infinity} b(n)/b(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}, b = A157257.

			7 - 2*sqrt(2) = 4.17157287525380990239...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Pierre-Antoine Guihéneuf, Rotations Discrètes, Images des Mathématiques, CNRS, 2018. See 3-2*sqrt(2), the fractional part of this constant, about the loss of information when rotating an image.
Index entries for algebraic numbers, degree 2

Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

[7 - 2*Sqrt(2)]; // G. C. Greubel, Nov 28 2017

RealDigits[7-2*Sqrt[2],10,120][[1]] (* Harvey P. Dale, May 01 2012 *)

7 - 2*sqrt(2) \\ G. C. Greubel, Nov 28 2017

Equals 3 + Sum_{k>=0} binomial(2*k,k)/((k+1) * 8^k). - Amiram Eldar, Aug 03 2020
Equals 4 + exp(-arccosh(3)). - Amiram Eldar, Jul 06 2023
Equals 4 + (2-sqrt(2))/(2+sqrt(2)). - Davide Rotondo, Jun 08 2024

A163960 Decimal expansion of 2*(sqrt(2) - 1).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 02 2010

Decimal expansion of shortest length, (B), of segment from side BC through incenter to side BA in right triangle ABC with sidelengths (a,b,c)=(1,1,sqrt(2)). (See A195284.) - Clark Kimberling, Sep 14 2011

			0.82842712474619009760337744841939615713934375075389614635335...

J. M. Steele, Probability Theory and Combinatorial Optimization, SIAM, 1997, p. 3.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Index entries for algebraic numbers, degree 2.

Cf. A084158, A156035, A195284.

Essentially the same digit sequence as A010466, A086178, A090488 and A157258.

RealDigits[2(Sqrt[2]-1),10,120][[1]] (* Harvey P. Dale, May 27 2016 *)

2*(sqrt(2)-1) \\ G. C. Greubel, Aug 13 2017

Equals Sum_{k>=0} (-1)^k * binomial(2*k,k)/((k+1) * 4^k). - Amiram Eldar, May 06 2022

Equals Sum_{k>=1} (-1)^(k+1)/A084158(k). - Amiram Eldar, Dec 02 2024

A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

Original entry on oeis.org

29, 41, 85, 89, 205, 481, 505, 1189, 2801, 2941, 6929, 16325, 17141, 40385, 95149, 99905, 235381, 554569, 582289, 1371901, 3232265, 3393829, 7996025, 18839021, 19780685, 46604249, 109801861, 115290281, 271629469, 639972145, 671961001

Offset: 1

Klaus Brockhaus, Feb 26 2009

(-20, a(1)) and (A129288(n), a(n+1)) are solutions (x, y) to the Diophantine equation x^2+(x+41)^2 = y^2.
lim_{n -> infinity} a(n)/a(n-3) = 3+2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (7+2*sqrt(2))/(7-2*sqrt(2)) for n mod 3 = {0, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (3+2*sqrt(2))*(7-2*sqrt(2))^2/(7+2*sqrt(2))^2 for n mod 3 = 1.

			(-20, a(1)) = (-20, 29) is a solution: (-20)^2+(-20+41)^2 = 400+441 = 841 = 29^2.
(A129288(1), a(2)) = (0, 41) is a solution: 0^2+(0+41)^2 = 1681 = 41^2.
(A129288(3), a(4)) = (39, 89) is a solution: 39^2+(39+41)^2 = 1521+6400 = 7921 = 89^2.

G. C. Greubel, Table of n, a(n) for n = 1..1001
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-1).

Cf. A129288, A001653, A156035 (decimal expansion of 3+2*sqrt(2)), A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)), A157260 (decimal expansion of (7+2*sqrt(2))/(7-2*sqrt(2))).

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6))) // G. C. Greubel, Feb 04 2018

CoefficientList[Series[(1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6), {x,0,50}], x] (* G. C. Greubel, Feb 04 2018 *)

{forstep(n=-20, 500000000, [3 ,1], if(issquare(n^2+(n+41)^2, &k), print1(k, ",")))}

x='x+O('x^30); Vec((1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+ x^6)) \\ G. C. Greubel, Feb 04 2018

a(n) = 6*a(n-3) - a(n-6) for n > 6; a(1)=29, a(2)=41, a(3)=85, a(4)=89, a(5)=205, a(6)=481.
G.f.: (1-x)*(29+70*x+155*x^2+70*x^3+29*x^4)/(1-6*x^3+x^6).
a(3*k-1) = 41*A001653(k) for k >= 1.

A157260 Decimal expansion of (7 + 2sqrt(2))/(7 - 2sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Feb 26 2009

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {1, 2}, b = A129288.

Equals lim_{n -> infinity} b(n)/b(n-1) for n mod 3 = {0, 2}, b = A157257.

			(7 +2*sqrt(2))/(7 -2*sqrt(2)) = 2.35604828649869905771...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for algebraic numbers, degree 2.

Cf. A129288, A157257, A157258 (decimal expansion of 7+2*sqrt(2)), A157259 (decimal expansion of 7-2*sqrt(2)).

Cf. A157300 (decimal expansion of (1683+58*sqrt(2))/41^2). - Klaus Brockhaus, May 01 2009

[(7+2*Sqrt(2))/(7-2*Sqrt(2))]; // G. C. Greubel, Nov 28 2017

RealDigits[(7 + 2*Sqrt[2])/(7 - 2*Sqrt[2]), 10, 50][[1]] (* G. C. Greubel, Nov 28 2017 *)

(7+2*sqrt(2))/(7-2*sqrt(2)) \\ G. C. Greubel, Nov 28 2017

Equals (57 + 28*sqrt(2))/41. - Klaus Brockhaus, May 01 2009

A365823 Decimal expansion of 2*(2 + sqrt(2)).

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Nov 13 2023

The greater one of the solutions to x^2 - 8 * x + 8 = 0. The other solution is A157259 - 3 = 1.17157... . - Michal Paulovic, Nov 14 2023

			6.8284271247461900976033774484193961571393437507538961...

Cf. A000129, A002193, A014176, A049310, A057084, A157259.
Cf. A156035, A163960.
Essentially the same as A157258, A090488, A086178 and A010466.

evalf(4+sqrt(8), 130);  # Alois P. Heinz, Nov 13 2023

First[RealDigits[2*(2 + Sqrt[2]), 10, 99]] (* Stefano Spezia, Nov 11 2023 *)

\\ Works in v2.13 and higher; n = 100 decimal places
my(n=100); digits(floor(10^n*(4+quadgen(32)))) \\ Michal Paulovic, Nov 14 2023

Equals 2*sqrt(2)*(1 + sqrt(2)) = 2*(2 + sqrt(2)). This is an integer in the quadratic number field Q(sqrt(2)).
Equals lim_{n->oo} A057084(n + 1)/A057084(n).
Equals continued fraction with periodic term [[6], [1, 4]]. - Peter Luschny, Nov 13 2023
Equals -3+A157258 = 1+A156035 = 2+A090488 = 3+A086178 = 4+A010466 = 6+A163960. - Alois P. Heinz, Nov 15 2023

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A090488 Decimal expansion of 2 + 2*sqrt(2).

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A129288 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 41)^2 = y^2.

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A157259 Decimal expansion of 7 - 2*sqrt(2).

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A163960 Decimal expansion of 2*(sqrt(2) - 1).

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A157257 Positive numbers y such that y^2 is of the form x^2+(x+41)^2 with integer x.

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A157260 Decimal expansion of (7 + 2*sqrt(2))/(7 - 2*sqrt(2)).

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A157260 Decimal expansion of (7 + 2sqrt(2))/(7 - 2sqrt(2)).