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Decimal expansion of the real root of x^3 - x^2 - 2 = 0.
+10
6
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
REFERENCES
D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves, unpublished, 1976, end of section 2. See links in A003229.
FORMULA
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
EXAMPLE
1.6956207695598620574163671001175353426181793882085077...
MATHEMATICA
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 *)
PROG
(PARI) solve(x=1, 2, x^3 - x^2 - 2) \\ Michel Marcus, Oct 26 2019
CROSSREFS
Sequences growing as this power: A003229, A003476, A003479, A052537, A077949, A144181, A164395, A164399, A164410, A164414, A164471, A203175, A227036, A289260, A292764.
Expansion of 1/((1-x)*(1-x-2*x^3)).
(Formerly M0781)
+10
4
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
REFERENCES
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 Unpublished, 1976.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).
MATHEMATICA
CoefficientList[Series[1/((1-x)*(1-x-2*x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 12 2012 *)
PROG
(PARI) 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
EXTENSIONS
More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 29 2003
Expansion of (x + x^2 + x^3) / (1 - x + x^2 - x^3 + x^4) in powers of x.
+10
2
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
FORMULA
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)).
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.
EXAMPLE
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 + ...
MATHEMATICA
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 *)
PROG
(PARI) {a(n) = [0, 1, 2, 2, 1][n%5 + 1] * (-1)^(n\5)};
(PARI) 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 (Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x)*(1-x^3)/((1-x)*(1+x^5)))); // G. C. Greubel, Aug 02 2018
Expansion of (1-x)/(1+x+2*x^3).
+10
1
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
CROSSREFS
Equals 4 * (-1)^n * A003476(n-2), n>2.
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