A155465 -id:A155465 - cp's OEIS frontend

A156035 Decimal expansion of 3 + 2*sqrt(2).

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

Offset: 1

Klaus Brockhaus, Feb 02 2009

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

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

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

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

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).

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

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

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

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

A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

Original entry on oeis.org

0, 7, 28, 51, 88, 207, 340, 555, 1248, 2023, 3276, 7315, 11832, 19135, 42676, 69003, 111568, 248775, 402220, 650307, 1450008, 2344351, 3790308, 8451307, 13663920, 22091575, 49257868, 79639203, 128759176, 287095935, 464171332, 750463515, 1673317776

Offset: 0

Mohamed Bouhamida, May 12 2006

Also values x of Pythagorean triples (x, x+17, y).
Corresponding values y of solutions (x, y) are in A155923.
For the generic case x^2+(x+p)^2 = y^2 with p = 2*m^2-1 a prime number in A066436, m >= 2 the associated value in A066049, the x values are given by the sequence defined by a(n) = 6*a(n-3)-a(n-6)+2p with a(0)=0, a(1)=2m+1, a(2)=6m^2-10m+4, a(3)=3p, a(4)=6m^2+10m+4, a(5)=40m^2-58m+21 (cf. A118673).
For the generic case x^2+(x+p)^2=y^2 with p=2*m^2-1 a prime number in A066436, m>=2, Y values are given by the sequence defined by: b(n)=6*b(n-3)-b(n-6) with b(0)=p, b(1)=2*m^2+2m+1, b(2)=10m^2-14m+5, b(3)=5p, b(4)=10m^2+14m+5, b(5)=58m^2-82m+29. - Mohamed Bouhamida, Sep 09 2009

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

Cf. A155923, A118673, A066436 (primes of the form 2*n^2-1), A066049 (2*n^2-1 is prime), A118554, A118611, A118630.

Cf. A155464 (first trisection), A155465 (second trisection), A155466 (third trisection).

[ n: n in [0..25000000] | IsSquare(2*n*(n+17)+289) ];

Select[Range[0,100000],IntegerQ[Sqrt[#^2+(#+17)^2]]&] (* or *) LinearRecurrence[{1,0,6,-6,0,-1,1},{0,7,28,51,88,207,340},50] (* Vladimir Joseph Stephan Orlovsky, Feb 02 2012 *)

m=32; v=concat([0, 7, 28, 51, 88, 207], vector(m-6)); for(n=7, m, v[n]=6*v[n-3]-v[n-6]+34); v

a(n) = 6*a(n-3) -a(n-6) +34 for n > 5; a(0)=0, a(1)=7, a(2)=28, a(3)=51, a(4)=88, a(5)=207.

G. f.: x*(7 +21*x +23*x^2 -5*x^3 -7*x^4 -5*x^5)/((1-x)*(1-6*x^3+x^6)).

Edited and 248755 changed to 248775 by Klaus Brockhaus, Feb 01 2009

A155464 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

Original entry on oeis.org

0, 51, 340, 2023, 11832, 69003, 402220, 2344351, 13663920, 79639203, 464171332, 2705388823, 15768161640, 91903581051, 535653324700, 3122016367183, 18196444878432, 106056652903443, 618143472542260, 3602804182350151

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

First trisection of A118120. Equals 17*A001652.

Cf. A155465, A155466, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[0,51,340]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Aug 21 2018

LinearRecurrence[{7,-7,1},{0,51,340},30] (* Harvey P. Dale, Jun 10 2013 *)
Table[17*(LucasL[2*n+1,2] - 2)/4, {n, 0, 50}] (* G. C. Greubel, Aug 21 2018 *)

{m=20; v=concat([0, 51, 340], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 0, a(1) = 51.
a(n) = ((1+sqrt(2))*(3+2*sqrt(2))^n + (1-sqrt(2))*(3-2*sqrt(2))^n -2)*(17/4).
G.f.: 17*x*(3-x)/((1-x)*(1-6*x+x^2)).
a(n) = 17*(A002203(2*n+1) - 2)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

A155466 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.

Original entry on oeis.org

28, 207, 1248, 7315, 42676, 248775, 1450008, 8451307, 49257868, 287095935, 1673317776, 9752810755, 56843546788, 331308470007, 1931007273288, 11254735169755, 65597403745276, 382329687301935, 2228380720066368

Offset: 0

Klaus Brockhaus, Jan 30 2009

lim_{n -> infinity} a(n+1)/a(n) = 3+2*sqrt(2).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (7,-7,1).

Third trisection of A118120. Cf. A001652.

Cf. A155464, A155465, A156035 (decimal expansion of 3+2*sqrt(2)).

I:=[28, 207, 1248]; [n le 3 select I[n] else 7*Self(n-1) - 7*Self(n-2) + Self(n-3): n in [1..50]]; // G. C. Greubel, Aug 21 2018

Table[(10*LucasL[2*n+3,2] + 3*LucasL[2*n+1, 2] -34)/4, {n, 0, 50}] (* or *) LinearRecurrence[{7,-7,1}, {28, 207, 1248}, 50] (* G. C. Greubel, Aug 21 2018 *)

{m=19; v=concat([28, 207, 1248], vector(m-3)); for(n=4, m, v[n]=7*v[n-1]-7*v[n-2]+v[n-3]); v}

a(n) = 6*a(n-1) - a(n-2) + 34 for n > 1; a(0) = 28, a(1) = 207.
a(n) = ((73+53*sqrt(2))*(3+2*sqrt(2))^n + (73-53*sqrt(2))*(3-2*sqrt(2))^n - 34)/4.
G.f.: (28+11*x-5*x^2)/((1-x)*(1-6*x+x^2)).
a(n) = (10*A002203(2*n+3) + 3*A002203(2*n+1) - 34)/4. - G. C. Greubel, Aug 21 2018

Comment and recursion formula added, cross-references edited by Klaus Brockhaus, Sep 23 2009

cp's OEIS Frontend

A156035 Decimal expansion of 3 + 2*sqrt(2).

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A118120 Nonnegative values x of solutions (x, y) to the Diophantine equation x^2+(x+17)^2 = y^2.

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A155464 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

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A155466 a(n) = 7*a(n-1) - 7*a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.

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Magma

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Extensions

A155464 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 0, a(1) = 51, a(2) = 340.

A155466 a(n) = 7a(n-1) - 7a(n-2) + a(n-3) for n > 2; a(0) = 28, a(1) = 207, a(2) = 1248.