login
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
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
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
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