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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
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OFFSET
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1,1
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COMMENTS
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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)
a(n) = A(n;1), where A(n;d), d in C, is the sequence of polynomials defined in Witula's comments to A189235 (see also Witula-Slota's paper for compatible sequences). - Roman Witula, Jul 26 2012
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REFERENCES
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R. Witula, D. Slota, Quasi-Fibonacci Numbers of Order 11, 10 (2007), Article 07.8.5.
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LINKS
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FORMULA
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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). - 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, Apr 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, Apr 20 2011
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MAPLE
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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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MATHEMATICA
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a[n_] := (1 - 2*Cos[Pi/11])^n + (2*Cos[(2*Pi)/11] + 1)^n + (1 - 2*Sin[Pi/22])^n + (2*Sin[(3*Pi)/22] + 1)^n + (1 - 2*Sin[(5*Pi)/22])^n; Table[a[n] // FullSimplify, {n, 1, 26}] (* Jean-François Alcover, Mar 26 2013 *)
u = {{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}}; a[n_] := Tr[MatrixPower[u, n]]; Table[a[n], {n, 1, 26}] (* Jean-François Alcover, Oct 16 2013, after L. Edson Jeffery *)
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PROG
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(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)) ) } \\ Harry J. Smith, Aug 12 2009
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009
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STATUS
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approved
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