A242412 -id:A242412

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A156035

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

+10
136

5, 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, 7, 0, 0, 2

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Limit_{n -> oo} b(n+1)/b(n) = 3+2*sqrt(2) for b = A155464, A155465, A155466.

Limit_{n -> oo} b(n)/b(n-1) = 3+2*sqrt(2) for b = A001652, A001653, A002315, A156156, A156157, A156158. - Klaus Brockhaus, Sep 23 2009

From Richard R. Forberg, Aug 14 2013: (Start)

Ratios b(n+1)/b(n) for all sequences of the form b(n) = 6*b(n-1) - b(n-2), for any initial values of b(0) and b(1), converge to this ratio.

Ratios b(n+1)/b(n) for all sequences of the form b(n) = 5*b(n-1) + 5*b(n-2) + b(n-3), for all b(0), b(1) and b(2) also converge to 3 + 2*sqrt(2). For example see A084158 (Pell Triangles).

Ratios of alternating values, b(n+2)/b(n), for all sequences of the form b(n) = 2*b(n-1) + b(n-2), also converge to 3 + 2*sqrt(2). These include A000129 (Pell Numbers). Also see A014176. (End)

Let ABCD be a square inscribed in a circle. When P is the midpoint of the arc AB, then the ratio (PC*PD)/(PA*PB) is equal to 3+2*sqrt(2). See the Mathematical Reflections link. - Michel Marcus, Jan 10 2017

Limit of ratios of successive terms of A001652 when n-> infinity. - Harvey P. Dale, Jun 16 2017; improved by Bernard Schott, Feb 28 2022

A quadratic integer with minimal polynomial x^2 - 6x + 1. - Charles R Greathouse IV, Jul 11 2020

Ratio between radii of the large circumscribed circle R and the small internal circle r drawn on the Sangaku tablet at Isaniwa Jinjya shrine in Ehime Prefecture (pictures in links). - Bernard Schott, Feb 25 2022

REFERENCES

Diogo Queiros-Condé and Michel Feidt, Fractal and Trans-scale Nature of Entropy, Iste Press and Elsevier, 2018, page 45.

LINKS

Ivan Panchenko, Table of n, a(n) for n = 1..1000

Mathematical Reflections, Solution to Problem J286, Issue 1, 2014, p. 5.

Bernard Schott, Sangaku at Isaniwa Jinya, The six circles.

Terakoya Suzu, Sangaku (mathematics tablet) II, Sangaku at Isaniwa Jinya shrine.

Wikipedia, Sangaku.

Bernard Ycart, Les Sangakus, Sangaku du Temple Isaniwa Jinya (in French).

Index entries for algebraic numbers, degree

FORMULA

Equals 1 + A090488 = 3 + A010466. - R. J. Mathar, Feb 19 2009

Equals exp(arccosh(3)), since arccosh(x) = log(x+sqrt(x^2-1)). - Stanislav Sykora, Nov 01 2013

Equals (1+sqrt(2))^2, that is, A014176^2. - Michel Marcus, May 08 2016

The periodic continued fraction is [5; [1, 4]]. - Stefano Spezia, Mar 17 2024

EXAMPLE

3 + 2*sqrt(2) = 5.828427124746190097603377448...

MATHEMATICA

RealDigits[N[3+2*Sqrt[2], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2011 *)

PROG

(PARI) 3+sqrt(8) \\ Charles R Greathouse IV, Jun 10 2011

(Magma) SetDefaultRealField(RealField(100)); 3 + 2*Sqrt(2); // G. C. Greubel, Aug 21 2018

CROSSREFS

Cf. A002193 (sqrt(2)), A090488, A010466, A014176.

Cf. A155464, A155465, A155466.

Cf. A104178 (decimal expansion of log_10(3+2*sqrt(2))).

Cf. A000129, A001109, A001541, A001542, A001652, A001653, A002315, A005319, A075870, A038723, A038725, A038761, A054488, A054489, A075848, A077413, A084158, A106328, A156156, A156157, A156158.

Cf. A242412 (sangaku).

KEYWORD

cons,easy,nonn

AUTHOR

Klaus Brockhaus, Feb 02 2009

STATUS

approved

A259555

a(n) = 2*n^2 - 2*n + 17.

+10
5

17, 21, 29, 41, 57, 77, 101, 129, 161, 197, 237, 281, 329, 381, 437, 497, 561, 629, 701, 777, 857, 941, 1029, 1121, 1217, 1317, 1421, 1529, 1641, 1757, 1877, 2001, 2129, 2261, 2397, 2537, 2681, 2829, 2981, 3137, 3297, 3461, 3629, 3801, 3977, 4157, 4341, 4529

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

a(n) is the curvature of the n-th touching circle in the area below the counterclockwise Pappus chain and the left semicircle of the arbelos with radii r0 = 2/3, r1 = 1/3. See illustration in the links.

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Kival Ngaokrajang, Illustration of initial terms

Eric Weisstein's World of Mathematics, Descartes Circle theorem.

Eric Weisstein's World of Mathematics, Pappus chain.

Wikipedia, Descartes' Theorem.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = 2*n^2 - 2*n + 17.

Descartes three circle theorem: a(n) = 3/2 + c(n) + c(n-1) + 2*sqrt(3*(c(n)+c(n-1))/2 + c(n)*c(n-1)), with c(n) = A114949(n)/2 = (n^2 + 6)/2, producing 2*n^2 - 2*n + 17. - Wolfdieter Lang, Jun 30 2015

From Colin Barker, Jul 01 2015: (Start)

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: -x*(17*x^2 - 30*x + 17)/(x-1)^3. (End)

E.g.f.: exp(x)*(2*x^2 + 17) - 17. - Elmo R. Oliveira, Nov 17 2024

MATHEMATICA

Table[2*n^2 - 2*n + 17, {n, 50}] (* Wesley Ivan Hurt, Feb 04 2017 *)

LinearRecurrence[{3, -3, 1}, {17, 21, 29}, 50] (* Harvey P. Dale, Apr 28 2017 *)

PROG

(PARI) a(n)=2*n^2-2*n+17

for (n=1, 100, print1(a(n), ", "))

(PARI) Vec(-x*(17*x^2-30*x+17)/(x-1)^3 + O(x^100)) \\ Colin Barker, Jul 01 2015

CROSSREFS

Cf. A114949, A242412 (for r0 = 1/2 = r1).

KEYWORD

nonn,easy,changed

AUTHOR

Kival Ngaokrajang, Jun 30 2015

EXTENSIONS

Edited by Wolfdieter Lang, Jun 30 2015

STATUS

approved

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