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A062883
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(1-2*cos(1/11*Pi))^n+(1+2*cos(2/11*Pi))^n+(1-2*cos(3/11*Pi))^n+(1+2*cos(4/11*Pi))^n+(1-2*cos(5/11*Pi))^n.
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4
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4, 12, 25, 64, 159, 411, 1068, 2808, 7423, 19717, 52529, 140251, 375015, 1003770, 2688570, 7204696, 19313075, 51782613, 138861732, 372414289, 998851473, 2679146955, 7186319506, 19276417059, 51707411684, 138702360471
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Contribution by L. Edson Jeffery, April 20, 2011. (Start):
Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,2)=
(0 0 1 0 0)
(0 1 0 1 0)
(1 0 1 0 1)
(0 1 0 1 1)
(0 0 1 1 1).
Then a(n)=Trace(U^(n+1)). Evidently this is one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix U_(N,r) (0<r<floor(N/2)) and for which the closed-form expression for a(n) is derived from the eigenvalues of U_(N,r). (End)
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LINKS
| L. E. Jeffery, Unit-primitive matrices.
Harry J. Smith, Table of n, a(n) for n=1,...,200
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FORMULA
| G.f.: x*(4-4*x-15*x^2+8*x^3+5*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 11 2009]
-A062883 = series expansion of (5-8*x-15*x^2+4*x^3+4*x^4)/(1-2*x-5*x^2+2*x^3+4*x^4+x^5) at x=infinity. (See also A189236.) - L. Edson Jeffery, April 20, 2011.
Also, a(n)=Sum_{k=1..5} ((w_k)^2-1)^(n+1), w_k=2*(-1)^(k-1)*cos(k*Pi/11), in which the polynomials {(w_k)^2-1} give the spectrum of the matrix U_(11,2) above. - L. Edson Jeffery, April 20, 2011.
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MAPLE
| Digits := 1000:q := seq(floor(evalf((1-2*cos(1/11*Pi))^n+(1+2*cos(2/11*Pi))^n+(1-2*cos(3/11*Pi))^n+(1+2*cos(4/11*Pi))^n+(1-2*cos(5/11*Pi))^n)), n=1..50);
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PROG
| (PARI) { default(realprecision, 200); for (n=1, 200, a=(1 - 2*cos(1/11*Pi))^n + (1 + 2*cos(2/11*Pi))^n + (1 - 2*cos(3/11*Pi))^n + (1 + 2*cos(4/11*Pi))^n + (1 - 2*cos(5/11*Pi))^n; write("b062883.txt", n, " ", round(a)) ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Aug 12 2009]
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CROSSREFS
| Cf. A033304, A062882, A189236.
Sequence in context: A008264 A000297 A078618 * A008176 A009903 A008048
Adjacent sequences: A062880 A062881 A062882 * A062884 A062885 A062886
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KEYWORD
| easy,nonn,changed
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AUTHOR
| Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 27 2001
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EXTENSIONS
| G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009.
More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Mar 24 2002
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