A189235 Expansion of (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5)

5, 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, 372064319188

Offset

0,1

Comments

Same as A062883 preceded by 5.

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).

Evidently 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).

Formulae given below are special cases of general one's defined and discussed in Witula-Slota's paper. For example a(n) = A(n;1), where A(n;d) := Sum_{k=1..5} (1 + 2d*cos(2Pi*k/11))^n, n=0,1,..., d in C. - Roman Witula, Jul 26 2012

References

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

Links

Table of n, a(n) for n=0..27.

L. E. Jeffery, Unit-primitive matrices

Formula

G.f.: (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5).

a(n)=4*a(n-1)-2*a(n-2)-5*a(n-3)+2*a(n-4)+a(n-5), {a(m)}=5,4,12,25,64, m=0..4.

a(n)=Sum_{k=1..5} ((x_k)^2-1)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).

Mathematica

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, 0, 27}] (* Jean-François Alcover, Oct 14 2013 *)

Prog

(PARI) Vec((5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012

Crossrefs

Cf. A189234, A189236, A189237.

Sequence in context: A284551 A316671 A338157 * A019068 A215947 A226555

Adjacent sequences: A189232 A189233 A189234 * A189236 A189237 A189238

Keyword

nonn,easy

Author

L. Edson Jeffery, Apr 18 2011

Status

approved