A365823 -id:A365823 - cp's OEIS frontend

A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2).

1, 8, 56, 384, 2624, 17920, 122368, 835584, 5705728, 38961152, 266043392, 1816657920, 12404916224, 84706066432, 578409201664, 3949625081856, 26969727041536, 184160815677440, 1257528709087232, 8586943147278336

Offset: 0

Wolfdieter Lang, Aug 11 2000

From Kival Ngaokrajang, Dec 14 2014 (Start):
-2*a(n-1) is the irrational part of the integer in Q(sqrt 2) giving the length of a Levy C-curve variant L(n)=(2*(2- sqrt 2))^n at iteration step n. The length of this C-curve is an integer in the real quadratic number field Q(sqrt 2), namely L(n) = A(n)+B(n)*sqrt(2) with A(n) = A084130(n) and B(n) = -2*a(n-1). See the construction rule and the illustration in the links.
The fractal dimension of the Levy C-curve is 2, but for this modified case it is log(4)/log(2 + sqrt 2) = 1.1289527...
(End)
For lim_{n->oo} a(n+1)/a(n) = 2*(2 + sqrt(2)) = 6.82842... see A365823. - Wolfdieter Lang, Nov 15 2023

			The first pairs [A(n),B(n)] determining the length L(n) are : [1, 0], [4, -2], [24, -16], [160, -112], [1088, -768], [7424, -5248], [50688, -35840], [346112, -244736], [2363392, -1671168], [16138240, -11411456], ... _Kival Ngaokrajang_, Dec 14 2014

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. F. Horadam, Special properties of the sequence W_n(a,b; p,q), Fib. Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=8, q=-8.
W. Lang, On polynomials related to powers of the generating function of Catalan's numbers, Fib. Quart. 38 (2000) 408-419. Eqs.(38) and (45),lhs, m=8.
Kival Ngaokrajang, Illustration of construction rule and initial terms
Wikipedia, Lévy C curve
Index entries for sequences related to Chebyshev polynomials.
Index entries for linear recurrences with constant coefficients, signature (8,-8).

Cf. A001109, A002315, A049310, A084130, A109466, A251732, A251733, A365823.

Join[{a=1,b=8},Table[c=8*b-8*a;a=b;b=c,{n,60}]] (* Vladimir Joseph Stephan Orlovsky, Jan 19 2011 *)
LinearRecurrence[{8,-8},{1,8},30] (* Harvey P. Dale, Feb 07 2015 *)

x='x+O('x^50); Vec(1/(1-8*x+8*x^2)) \\ G. C. Greubel, Jul 03 2017

[lucas_number1(n,8,8) for n in range(1, 21)] # Zerinvary Lajos, Apr 23 2009

a(n) = 8*(a(n-1)-a(n-2)), a(-1)=0, a(0)=1.
a(n) = S(n, 2*sqrt(2))*(2*sqrt(2))^n with S(n, x) := U(n, x/2), Chebyshev's polynomials of the 2nd kind, A049310.
a(2*k) = A002315(k)*8^k, a(2*k+1) = A001109(k+1)*8^(k+1).
G.f.: 1/(1-8*x+8*x^2).
a(n) = Sum_{k, 0<=k<=n} A109466(n,k)*8^k. [Philippe Deléham, Oct 28 2008]
Binomial transform of A002315. [Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009]

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

A100954 Decimal expansion of 7/2 - sqrt(2)/4.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Jan 11 2005

An approximation to Pi, see Blatner. - Robert G. Wilson v, Dec 15 2012

This constant is three plus the ratio of the area (A) to the perimeter (P) of an isosceles right triangle with unit legs, 3 + A/P = 3 + (1/2)/(2 + sqrt(2)). - José de Jesús Camacho Medina, Mar 02 2018

			3.14644660940672623779957781894757548035758203115576298170583...

David Blatner, Spectrums, Our Mind-Boggling Universe from Infinitesimal to Infinity, Walker & Co, USA, 2012.

G. C. Greubel, Table of n, a(n) for n = 1..10000

SetDefaultRealField(RealField(100)); (14 - Sqrt(2))/4; // G. C. Greubel, Oct 01 2018

RealDigits[(14 - Sqrt[2])/4, 10, 111][[1]] (* Robert G. Wilson v, Dec 15 2012 *)

7/2 - sqrt(2)/4 \\ Charles R Greathouse IV, Mar 25 2014

Equals (14 - sqrt(2))/4 = 3 + 1/J, where J = 2*(2 + sqrt(2)) = A365823. - Wolfdieter Lang, Nov 11 2023

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A057084 Scaled Chebyshev U-polynomials evaluated at sqrt(2).

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

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A100954 Decimal expansion of 7/2 - sqrt(2)/4.

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