A003476 -id:A003476 - cp's OEIS frontend

A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

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

Offset: 1

Clark Kimberling, Jul 14 2017

			1.6956207695598620574163671001175353426181793882085077...

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.

Angel Chang and Tianrong Zhang, The Fractal Geometry of the Boundary of Dragon Curves, Journal of Recreational Mathematics, volume 30, number 1, 1999-2000, pages 9-22.
Index entries for algebraic numbers, degree 3.

Sequences growing as this power: A003229, A003476, A003479, A052537, A077949, A144181, A164395, A164399, A164410, A164414, A164471, A203175, A227036, A289260, A292764.

Cf. A078140 (includes guide to constants similar to A289260).

z = 2000; r = 8/5;
u = CoefficientList[Series[1/Sum[Floor[(k + 1)*r] (-x)^k, {k, 0, z}], {x, 0, z}], x];  (* A289260 *)
v = N[u[[z]]/u[[z - 1]], 200]
RealDigits[v, 10][[1]] (* A289265 *)

solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019

r = D^(1/3) + (1/9)*D^(-1/3) + 1/3 where D = 28/27 + (1/9)*sqrt(29*3) [Chang and Zhang] from the usual cubic solution formula. Or similarly r = (1/3)*(1 + C + 1/C) where C = (28 + sqrt(29*27))^(1/3). - Kevin Ryde, Oct 25 2019

A003479 Expansion of 1/((1-x)(1-x-2x^3)).

Original entry on oeis.org

1, 2, 3, 6, 11, 18, 31, 54, 91, 154, 263, 446, 755, 1282, 2175, 3686, 6251, 10602, 17975, 30478, 51683, 87634, 148591, 251958, 427227, 724410, 1228327, 2082782, 3531603, 5988258, 10153823, 17217030, 29193547, 49501194, 83935255, 142322350

Offset: 0

N. J. A. Sloane

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links in A003229 for an earlier version.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
D. E. Daykin, Letter to N. J. A. Sloane, Mar 1974.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
Index entries for linear recurrences with constant coefficients, signature (2,-1,2,-2).

Cf. A003229.

A003479:=1/(z-1)/(-1+z+2*z**3); # Simon Plouffe in his 1992 dissertation

CoefficientList[Series[1/((1-x)*(1-x-2*x^3)),{x,0,40}],x] (* Vincenzo Librandi, Jun 12 2012 *)

a(n)=([0,1,0,0; 0,0,1,0; 0,0,0,1; -2,2,-1,2]^n*[1;2;3;6])[1,1] \\ Charles R Greathouse IV, Jun 23 2020

A003476(n+1) + A077949(n)/2 - 1/2. - Ralf Stephan, Sep 25 2004

a(n+1) - a(n) = A077949(n+1). - R. J. Mathar, Mar 22 2011

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003

A253262 Expansion of (x + x^2 + x^3) / (1 - x + x^2 - x^3 + x^4) in powers of x.

Original entry on oeis.org

0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1, -2, -2, -1, 0, 1, 2, 2, 1, 0, -1

Offset: 0

Michael Somos, Apr 30 2015

Cycle period is 10. - Robert G. Wilson v, Aug 02 2018

			G.f. = x + 2*x^2 + 2*x^3 + x^4 - x^6 - 2*x^7 - 2*x^8 - x^9 + x^11 + 2*x^12 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1).

Cf. A003476.

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^3)/((1-x)*(1+x^5)))); // G. C. Greubel, Aug 02 2018

a[ n_] := {1, 2, 2, 1, 0}[[Mod[n, 5, 1]]] (-1)^Quotient[n, 5];
CoefficientList[Series[x*(1+x)*(1-x^3)/((1-x)*(1+x^5)), {x,0,60}], x] (* G. C. Greubel, Aug 02 2018 *)
CoefficientList[ Series[x (x^2 + x + 1)/(x^4 - x^3 + x^2 - x + 1), {x, 0, 75}], x] (* or *)
LinearRecurrence[{1, -1, 1, -1}, {0, 1, 2, 2}, 75] (* Robert G. Wilson v, Aug 02 2018 *)

{a(n) = [0, 1, 2, 2, 1][n%5 + 1] * (-1)^(n\5)};

x='x+O('x^60); concat([0], Vec(x*(1+x)*(1-x^3)/((1-x)*(1+x^5)))) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 10 sequence [2, -1, -1, 0, -1, 0, 0, 0, 0, 1].
G.f.: x * (1 + x) * (1 - x^3) / ((1 - x) * (1 + x^5)).
INVERT transform is A003476.
a(n) = -a(-n) = -a(n+5) for all n in Z.
a(n) = f(n) / f(1) where f(n) := tan( am( n*x, m)) where x = 0.7379409146... and m = 1.3481185591... and am() is the Jacobi amplitude function.

A078044 Expansion of (1-x)/(1+x+2*x^3).

Original entry on oeis.org

1, -2, 2, -4, 8, -12, 20, -36, 60, -100, 172, -292, 492, -836, 1420, -2404, 4076, -6916, 11724, -19876, 33708, -57156, 96908, -164324, 278636, -472452, 801100, -1358372, 2303276, -3905476, 6622220, -11228772, 19039724, -32284164, 54741708, -92821156, 157389484, -266872900

Offset: 0

N. J. A. Sloane, Nov 17 2002

Index entries for linear recurrences with constant coefficients, signature (-1,0,-2).

Equals 4 * (-1)^n * A003476(n-2), n>2.

First differences of A077974.

Vec((1-x)/(1+x+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 27 2012

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A289265 Decimal expansion of the real root of x^3 - x^2 - 2 = 0.

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A003479 Expansion of 1/((1-x)*(1-x-2*x^3)).

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A253262 Expansion of (x + x^2 + x^3) / (1 - x + x^2 - x^3 + x^4) in powers of x.

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A078044 Expansion of (1-x)/(1+x+2*x^3).

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A003479 Expansion of 1/((1-x)(1-x-2x^3)).