A002193 -id:A002193 - cp's OEIS frontend

A010487 Decimal expansion of square root of 32.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 5 followed by {1, 1, 1, 10} repeated. - Harry J. Smith, Jun 04 2009
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)=(5,12,13), see A195284. - Clark Kimberling, Sep 14 2011
Length of the perimeter of a square with circumscribed unit circle. - R. J. Mathar, Aug 24 2023

			5.656854249492380195206754896838792314278687501507792292706718951962929....

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A010130 (continued fraction).

RealDigits[N[Sqrt[32],200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 22 2011 *)

default(realprecision, 20080); x=sqrt(32); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010487.txt", n, " ", d));  \\ Harry J. Smith, Jun 04 2009

Equals 4 * A002193. - R. J. Mathar, Oct 16 2015

A182007 Decimal expansion of 2*sin(Pi/5).

Original entry on oeis.org

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

Offset: 1

Stanislav Sykora, Apr 06 2012

The golden ratio phi is the real part of 2*exp(i*Pi/5), while this constant c is the corresponding imaginary part. It is handy, for example, in simplifying metric expressions for Platonic solids (particularly for regular icosahedron and dodecahedron).
Note that c^2+A001622^2 = 4; c*A001622 = A188593 = 2*A019881; c = 2*A019845.
Edge length of a regular pentagon with unit circumradius. - Stanislav Sykora, May 07 2014
This is a constructible number (see A003401 for more details). Moreover, since phi is also constructible, (2^k)*exp(i*Pi/5), for any integer k, is a constructible complex number. - Stanislav Sykora, May 02 2016
rms(c, phi) := sqrt((c^2+phi^2)/2) = sqrt(2) = A002193.

			1.1755705045849462583374119...

Ivan Panchenko, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Pentagon.
Wikipedia, Platonic solid.
Index entries for algebraic numbers, degree 4.

Cf. A001622, A002193, A003401, A019881, A019845, A063226, A102769, A131595, A188593, A300074.

SetDefaultRealField(RealField(100)); R:= RealField(); 2*Sin(Pi(R)/5); // G. C. Greubel, Nov 02 2018

evalf(2*sin(Pi/5),100); # Muniru A Asiru, Nov 02 2018

RealDigits[2*Sin[Pi/5],10,120][[1]] (* Harvey P. Dale, Sep 29 2012 *)

2*sin(Pi/5) \\ Stanislav Sykora, May 02 2016

Equals sqrt(3-phi).
Equals sqrt((5-sqrt(5))/2). - Jean-François Alcover, May 21 2013
Equals Product_{k>=0} ((10*k + 4)*(10*k + 6))/((10*k + 3)*(10*k + 7)). - Antonio Graciá Llorente, Mar 25 2024
Equals Product_{k>=1} (1 - (-1)^k/A063226(k)). - Amiram Eldar, Nov 23 2024
Equals 2*A019845 = 1/A300074. - Hugo Pfoertner, Nov 23 2024

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

Original entry on oeis.org

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

A101465 Decimal expansion of 2-sqrt(2), square of the edge length of a regular octagon with circumradius 1.

Original entry on oeis.org

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

Offset: 0

Jun Mizuki (suzuki32(AT)sanken.osaka-u.ac.jp), Jan 20 2005

This also equals the probability, for a random walk on the slit plane starting at (1,0), of stopping at the origin.
This is the least real number m such that m*(sqrt(ab) + sqrt(bc) + sqrt(ca)) + sqrt(a^2+b^2+c^2) >= a+b+c where a,b,c are positive real numbers with ab+bc+ca > 0. See the Mathematical Reflections link.
The asymptotical ratio of odd to even powerful numbers (Srichan, 2020). - Amiram Eldar, Mar 07 2021
The volume of the solid formed by the intersection of 3 right circular unit-diameter cylinders whose axes are mutually orthogonal and intersect at a single point (Moore, 1974). - Amiram Eldar, Nov 22 2021
Smallest root of the Laguerre polynomial of degree 2. - A.H.M. Smeets, May 15 2025

			0.585786437626904951198311275790301921430328124623051926823320262...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Mireille Bousquet-Mélou and Gilles Schaeffer, Walks on the slit plane, Probability Theory and Related Fields, Vol. 124, No. 3 (2002), pp. 305-344; arXiv preprint, arXiv:0012230 [math.CO], 2000.
Mathematical Reflections, Solution to Problem J307, Issue 5, 2014, p. 21.
Moreton Moore, Symmetrical Intersections of Right Circular Cylinders, The Mathematical Gazette, Vol. 58, No. 405 (1974), pp. 181-185.
Teerapat Srichan, The Odd/Even Dichotomy For The Set Of Square-Full Numbers, Applied Mathematics E-Notes, Vol. 20 (2020), pp. 528-531.
Eric Weisstein's World of Mathematics, Octagon.
Index entries for algebraic numbers, degree 2.

Cf. A001694, A002193 (sqrt(2)).

SetDefaultRealField(RealField(100));  2 - Sqrt(2); // G. C. Greubel, Oct 01 2018

RealDigits[2 - Sqrt[2], 10, 100][[1]] (* G. C. Greubel, Oct 01 2018 *)

2 - sqrt(2) \\ Michel Marcus, Jan 07 2017

A257574 Continued square root map applied to the sequence of positive even numbers, (2, 4, 6, 8, ...).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 02 2015

The continued square root or CSR map applied to a sequence b = (b(1), b(2), b(3), ...) is the number CSR(b) := sqrt(b(1)+sqrt(b(2)+sqrt(b(3)+sqrt(b(4)+...)))).

Taking out a factor sqrt(2), one gets CSR(2, 4, 6, 8, ...) = sqrt(2) CSR(1, 1, 3/8, 1/32, ...) < A002193*A001622 = (sqrt(5)+1)/sqrt(2). - M. F. Hasler, May 01 2018

			sqrt(2 + sqrt(4 + sqrt(6 + sqrt(8 + ...)))) = 2.1584768723110397656558534...

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..400
A. Herschfeld, On Infinite Radicals, Amer. Math. Monthly, 42 (1935), 419-429.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Cf. A072449, A257575..A257581, A105817, A099879, A001622, A105546.

(CSR(v,s)=forstep(i=#v,1,-1,s=sqrt(v[i]+s));s); t=0;for(N=5,oo,(t==t=Str(CSR([1..2*N]*2)))&&break;print(2*N": "t)) \\ Allows to see the convergence, which is reached when length of vector ~ precision [given as number of digits]. Using Str() to avoid infinite loop when internal representation is "fluctuating". - M. F. Hasler, May 04 2018

a(27)-a(87) from Hiroaki Yamanouchi, May 03 2015

Edited by M. F. Hasler, May 01 2018

A091999 Numbers that are congruent to {2, 10} mod 12.

Original entry on oeis.org

2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98, 106, 110, 118, 122, 130, 134, 142, 146, 154, 158, 166, 170, 178, 182, 190, 194, 202, 206, 214, 218, 226, 230, 238, 242, 250, 254, 262, 266, 274, 278, 286, 290, 298, 302, 310, 314, 322, 326, 334

Offset: 1

Ray Chandler, Feb 21 2004

Numbers divisible by 2 but not by 3 or 4. - Robert Israel, Apr 24 2015
For n > 1, a(n) is representable as a sum of four but no fewer consecutive nonnegative integers, i.e., 10 = 1 + 2 + 3 + 4, 14 = 2 + 3 + 4 + 5, 22 = 4 + 5 + 6 + 7, etc. (see A138591). - Martin Renner, Mar 14 2016
Essentially the same as A063221. - Omar E. Pol, Aug 16 2023

David Lovler, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Second row of A092260.
Cf. A109761 (subsequence).
Cf. A002193, A007310, A101263, A138591, A186424.
Cf. A017545, A017641, A063221.

a091999 n = a091999_list !! (n-1)
a091999_list = 2 : 10 : map (+ 12) a091999_list
-- Reinhard Zumkeller, Jan 21 2013

[6*n-3+(-1)^n : n in [1..100]]; // Wesley Ivan Hurt, Apr 23 2015

A091999:=n->6*n-3+(-1)^n: seq(A091999(n), n=1..100); # Wesley Ivan Hurt, Apr 23 2015

Flatten[#+{2,10}&/@(12*Range[0,30])] (* or *) LinearRecurrence[{1,1,-1},{2,10,14},60] (* Harvey P. Dale, Jun 24 2013 *)

a(n) = 6*n - 3 + (-1)^n \\ David Lovler, Jul 16 2022

a(n) = 2*A007310(n).
a(n) = A186424(n) - A186424(n-2), for n > 1.
a(n) = 12*(n-1) - a(n-1), with a(1)=2. - Vincenzo Librandi, Nov 16 2010
G.f.: 2*x*(1+4*x+x^2) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Oct 08 2011
a(n) = a(n-1) + a(n-2) - a(n-3); a(1)=2, a(2)=10, a(3)=14. - Harvey P. Dale, Jun 24 2013
a(n) = 6*n - 3 + (-1)^n. - Wesley Ivan Hurt, Apr 23 2015
E.g.f.: 2 + (6*x - 2)*cosh(x) + 2*(3*x - 2)*sinh(x). - Stefano Spezia, May 09 2021
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(4*sqrt(3)). - Amiram Eldar, Dec 13 2021
E.g.f.: 2 + (6*x - 3)*exp(x) + exp(-x). - David Lovler, Aug 08 2022
a(n) = A063221(n), n > 1. - Omar E. Pol, Aug 15 2023
From Amiram Eldar, Nov 24 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sqrt(2) (A002193).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/12) (A101263). (End)

A010524 Decimal expansion of square root of 72.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

This is also the ratio of the volume of a cube to the volume of a regular tetrahedron of the same edge length. - Rick L. Shepherd, May 29 2002
Continued fraction expansion is 8 followed by {2, 16} repeated. - Harry J. Smith, Jun 08 2009
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)=(8,15,17), see A195284. - Clark Kimberling, Sep 14 2011
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)=(7,24,25). - Clark Kimberling, Sep 14 2011

			8.485281374238570292810132345258188471418031252261688439060078427944394...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Index entries for algebraic numbers, degree 2

Cf. A040063 (continued fraction), A002193, A195284.

RealDigits[N[72^(1/2),200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jan 23 2012 *)

default(realprecision, 20080); x=sqrt(72); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010524.txt", n, " ", d));  \\ Harry J. Smith, Jun 08 2009

Equals 6*A002193. - Omar E. Pol, Mar 09 2021

A179260 Decimal expansion of the connective constant of the honeycomb lattice.

Original entry on oeis.org

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

Offset: 1

Jonathan Vos Post, Jul 06 2010

This is the case n=8 of the ratio Gamma(1/n)*Gamma((n-1)/n)/(Gamma(2/n)*Gamma((n-2)/n)). - Bruno Berselli, Dec 13 2012
An algebraic integer of degree 4: largest root of x^4 - 4x^2 + 2. - Charles R Greathouse IV, Nov 05 2014
This number is also the length ratio of the shortest diagonal (not counting the side) of the octagon and the side. This ratio is A121601 for the longest diagonal. - Wolfdieter Lang, May 11 2017 [corrected Oct 28 2020]
From Wolfdieter Lang, Apr 29 2018: (Start)
This constant appears in a historic problem posed by Adriaan van Roomen (Adrianus Romanus) in his Ideae mathematicae from 1593, solved by Viète. See the Havil reference, problem 3, pp. 69-74. See also the comments in A302711 with the Romanus link and his Exemplum tertium.
This problem is equivalent to R(45, 2*sin(Pi/120)) = 2*sin(3*Pi/8) with a special case of monic Chebyshev polynomials of the first kind, named R, given in A127672. For the constant 2*sin(Pi/120) see A302715. (End)

			1.84775906502257351225636637879357657364483325172728497223019546256107001500...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.10, p. 333.
Julian Havil, The Irrationals, A Story of the Numbers You Can't Count On, Princeton University Press, Princeton and Oxford, 2012, pp. 69-74.
Neal Madras and Gordon Slade, Self-avoiding walks, Probability and its Applications, Birkhäuser Boston, Inc. Boston, MA, 1993.

Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), arXiv:1007.0575 [math-ph], 2011.
Hugo Duminil-Copin and Stanislav Smirnov, The connective constant of the honeycomb lattice equals sqrt(2+sqrt2), Ann. Math. 175 (2012), pp. 1653-1665.
Steven Finch, Errata and Addenda to Mathematical Constants, Jun 23 2012, Section 5.10; arXiv:2001.00578 [math.HO], 2020.
Pierre-Louis Giscard, Que sait-on compter sur un graphe. Partie 3 (in French), Images des Mathématiques, CNRS, 2020.
Gregory F. Lawler, Oded Schramm, and Wendelin Werner, On the scaling limit of planar self-avoiding walk, Fractal Geometry and applications: a jubilee of Benoit Mandelbrot, Part 2, 339-364. Proc.
Bernard Nienhuis, Exact critical point and critical exponents of O(n) models in two dimensions, Phys. Rev. Lett. 49 (1982), 1062-1065.
Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), arXiv:1005.2712 [math.NT], 2010.
Jonathan Sondow and Huang Yi, New Wallis- and Catalan-type infinite products for Pi, e, and sqrt(2+sqrt(2)), Amer. Math. Monthly 117 (2010) 912-917.
Index entries for algebraic numbers, degree 4.
Index entries for sequences related to Chebyshev polynomials.

Cf. A002193, A047522, A101464, A127672, A144981, A154739, A302715, A121601.

RealDigits[Sqrt[2+Sqrt[2]],10,120][[1]] (* Harvey P. Dale, Jan 19 2014 *)

sqrt(2+sqrt(2)) \\ Charles R Greathouse IV, Nov 05 2014

sqrt(2+sqrt(2)) = (2/1)(6/7)(10/9)(14/15)(18/17)(22/23)... (see Sondow-Yi 2010).
Equals 1/A154739. - R. J. Mathar, Jul 11 2010
Equals 2*A144981. - Paul Muljadi, Aug 23 2010
log (A001668(n)) ~ n log k where k = sqrt(2+sqrt(2)). - Charles R Greathouse IV, Nov 08 2013
2*cos(Pi/8) = sqrt(2+sqrt(2)). See a remark on the smallest diagonal in the octagon above. - Wolfdieter Lang, May 11 2017
Equals also 2*sin(3*Pi/8). See the comment on van Roomen's third problem above. - Wolfdieter Lang, Apr 29 2018
Equals i^(1/4) + i^(-1/4). - Gary W. Adamson, Jul 06 2022
Equals Product_{k>=0} ((8*k + 2)*(8*k + 6))/((8*k + 1)*(8*k + 7)). - Antonio Graciá Llorente, Feb 24 2024
Equals Product_{k>=1} (1 - (-1)^k/A047522(k)). - Amiram Eldar, Nov 22 2024

A156164 Decimal expansion of 17 + 12*sqrt(2).

Original entry on oeis.org

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

Offset: 2

Klaus Brockhaus, Feb 09 2009

Lim_{n -> infinity} b(n)/b(n-1) = 17+12*sqrt(2) for b = A156160, A156161, A156162, A278310.

Conjecturally, the fractional part 0.97056 27484 ... of this constant equals ( (1 + 2 * Sum_{n >= 1} (-1)^n*exp(-2*Pi*n^2))/(1 + 2 * Sum_{n >= 1} exp(-2*Pi*n^2)) )^4. The series are rapidly converging. For example, summing both series from n = 1 to n = 2 approximates the fractional part of the constant as ( (1 - 2*exp(-2*Pi) + 2*exp(-8*Pi))/(1 + 2*exp(-2*Pi) + 2*exp(-8*Pi)) )^4 = 0.97056 27484 77140 58562 026(89) ..., correct to 23 decimal places. - Peter Bala, Jun 05 2019

			17 + 12*sqrt(2) = 33.97056274847714058562026469051637694283606250452337687...

G. C. Greubel, Table of n, a(n) for n = 2..10000
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.4. Example 4.
Index entries for algebraic numbers, degree 2

Cf. A002193: decimal expansion of sqrt(2); A156035: decimal expansion of 3+2*sqrt(2); A156163: decimal expansion of (19+6*sqrt(2))/17.

Cf. A005259, A059415/A059416.

RealDigits[17 + 12 Sqrt[2], 10, 120][[1]] (* Harvey P. Dale, Nov 30 2014 *)

17+sqrt(288) \\ Charles R Greathouse IV, May 07 2015

17+12*sqrt(2) = (3+2*sqrt(2))^2 = (1+sqrt(2))^4. - Klaus Brockhaus, Feb 14 2009. (corrected by Bruno Berselli, Feb 19 2013)

A118674 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2 + (x + 31)^2 = y^2.

Original entry on oeis.org

0, 9, 60, 93, 140, 429, 620, 893, 2576, 3689, 5280, 15089, 21576, 30849, 88020, 125829, 179876, 513093, 733460, 1048469, 2990600, 4274993, 6111000, 17430569, 24916560, 35617593, 101592876, 145224429, 207594620, 592126749, 846430076

Offset: 1

Mohamed Bouhamida, May 19 2006

Also values x of Pythagorean triples (x, x+31, y).
Corresponding values y of solutions (x, y) are in A157646.
For the generic case x^2 + (x + p)^2 = y^2 with p = 2*m^2 - 1 a (prime) number in A066436 see A118673 or A129836.
lim_{n -> infinity} a(n)/a(n-3) = 3 + 2*sqrt(2).
lim_{n -> infinity} a(n)/a(n-1) = (33 + 8*sqrt(2))/31 for n mod 3 = {1, 2}.
lim_{n -> infinity} a(n)/a(n-1) = (1539 + 850*sqrt(2))/31^2 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. A157646, A066436 (primes of the form 2*n^2-1), A118673, A129836, A001652, A002193 (decimal expansion of sqrt(2)), A156035 (decimal expansion of 3 + 2*sqrt(2)), A157647 (decimal expansion of (33 + 8*sqrt(2))/31), A157648 (decimal expansion of (1539 + 850*sqrt(2))/31^2).

I:=[0,9,60,93,140,429,620]; [n le 7 select I[n] else Self(n-1) - 6*Self(n-3) - 6*Self(n-4) - Self(n-6) + Self(n-7): n in [1..50]]; // G. C. Greubel, Mar 31 2018

ClearAll[a]; Evaluate[Array[a, 6]] = {0, 9, 60, 93, 140, 429}; a[n_] := a[n] = 6*a[n-3] - a[n-6] + 62; Table[a[n], {n, 1, 31}] (* Jean-François Alcover, Dec 27 2011, after given formula *)
LinearRecurrence[{1,0,6,-6,0,-1,1}, {0,9,60,93,140,429,620}, 50] (* G. C. Greubel, Mar 31 2018 *)

{forstep(n=0, 850000000, [1, 3], if(issquare(2*n^2+62*n+961), print1(n, ",")))};

a(n) = 6*a(n-3) - a(n-6) + 62 for n > 6; a(1)=0, a(2)=9, a(3)=60, a(4)=93, a(5)=140, a(6)=429.
G.f.: x*(9 + 51*x + 33*x^2 - 7*x^3 - 17*x^4 - 7*x^5)/((1-x)*(1 - 6*x^3 + x^6)).
a(3*k + 1) = 31*A001652(k) for k >= 0.

Edited by Klaus Brockhaus, Mar 11 2009

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