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A062883 (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. 4
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; text; internal format)
OFFSET

1,1

COMMENTS

From L. Edson Jeffery, Apr 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)

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

REFERENCES

R. Witula, D. Slota, Quasi-Fibonacci Numbers of Order 11, 10 (2007), Article 07.8.5.

LINKS

Harry J. Smith, Table of n, a(n) for n=1,...,200

L. E. Jeffery, Unit-primitive matrices

R. Wituła, D. Słota, Quasi-Fibonacci Numbers of Order 11, Journal of Integer Sequences, Vol. 10 (2007), Article 07.8.5

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). - 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

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);

MATHEMATICA

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 *)

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)) ) } \\ Harry J. Smith, Aug 12 2009

CROSSREFS

Cf. A033304, A062882, A189236.

Sequence in context: A008264 A000297 A078618 * A008176 A009903 A008048

Adjacent sequences:  A062880 A062881 A062882 * A062884 A062885 A062886

KEYWORD

easy,nonn

AUTHOR

Vladeta Jovovic, Jun 27 2001

EXTENSIONS

G.f. proposed by Maksym Voznyy checked and corrected by R. J. Mathar, Sep 16 2009

More terms from Sascha Kurz, Mar 24 2002

STATUS

approved

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Last modified February 24 13:22 EST 2018. Contains 299623 sequences. (Running on oeis4.)