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
A189236 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).
5, 2, 14, 32, 114, 347, 1142, 3649, 11826, 38111, 123139, 397443, 1283406, 4143479, 13378435, 43194542, 139463234, 450284986, 1453839839, 4694021537, 15155624819, 48933074467, 157990585613, 510105367936, 1646980994190, 5317619734147
Offset: 0
Comments
(Start) Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,3)=
(0 0 0 1 0)
(0 0 1 0 1)
(0 1 0 1 1)
(1 0 1 1 1)
(0 1 1 1 1).
Then a(n)=Trace(U^n). (End)
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
Links
- L. E. Jeffery, Unit-primitive matrices
- Index entries for linear recurrences with constant coefficients, signature (2, 5, -2, -4, -1).
Programs
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Mathematica
CoefficientList[Series[ (5-8x-15x^2+4x^3+4x^4)/ (1-2x-5x^2+2x^3+4x^4+x^5), {x,0,29}],x] (* Harvey P. Dale, Apr 19 2011 *) LinearRecurrence[{2, 5, -2, -4, -1}, {5, 2, 14, 32, 114}, 30] (* T. D. Noe, Apr 19 2011 *) -
PARI
Vec((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)+O(x^99)) \\ Charles R Greathouse IV, Sep 25 2012
Formula
G.f.: (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).
a(n)=2*a(n-1)+5*a(n-2)-2*a(n-3)-4*a(n-4)-a(n-5), {a(m)}={5,2,14,32,114}, m=0..4.
a(n)=Sum_{k=1..5} ((x_k)^3-2*(x_k))^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).
A062882 a(n) = (1 - 2*cos(Pi/9))^n + (1 + 2*cos(Pi*2/9))^n + (1 + 2*cos(Pi*4/9))^n.
3, 9, 18, 45, 108, 270, 675, 1701, 4293, 10854, 27459, 69498, 175932, 445419, 1127763, 2855493, 7230222, 18307377, 46355652, 117376290, 297206739, 752553261, 1905530913, 4824972522, 12217257783, 30935180610, 78330624264
Offset: 1
Comments
From L. Edson Jeffery, Apr 05 2011: (Start)
Let U be the matrix (see [Jeffery])
U = U_(9,2) =
(0 0 1 0)
(0 1 0 1)
(1 0 1 1)
(0 1 1 1).
Then a(n) = Trace(U^n).
(End)
We note that all numbers of the form a(n)*3^(-floor((n+4)/3)) are integers. - Roman Witula, Sep 29 2012
Examples
We have a(2)=3*a(1), a(4)/a(3) = a(6)/a(5) = a(7)/a(6) = 5/2, a(6)=6*a(4), a(7)=15*a(4), and (1 + c(1))^8 + (1 + c(2))^8 + (1 + c(4))^8 = 7*3^5. - _Roman Witula_, Sep 29 2012
Links
- Harry J. Smith, Table of n, a(n) for n = 1..200
- L. Edson Jeffery, Danzer matrices
- Index entries for linear recurrences with constant coefficients, signature (3,0,-3).
Programs
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Maple
Digits := 1000:q := seq(floor(evalf((1-2*cos(1/9*Pi))^n+(1+2*cos(2/9*Pi))^n+(1+2*cos(4/9*Pi))^n)),n=1..50);
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Mathematica
LinearRecurrence[{3,0,-3},{3,9,18},25] (* Georg Fischer Feb 02 2019 *) -
PARI
{ default(realprecision, 200); for (n=1, 200, a=(1 - 2*cos(1/9*Pi))^n + (1 + 2*cos(2/9*Pi))^n + (1 + 2*cos(4/9*Pi))^n; write("b062882.txt", n, " ", round(a)) ) } \\ Harry J. Smith, Aug 12 2009 -
PARI
Vec((3-9*x^2)/(1-3*x+3*x^3)+O(x^66)) /* Joerg Arndt, Apr 08 2011 */
Formula
G.f.: x*(3 - 9*x^2)/(1 - 3*x + 3*x^3). The terms in parentheses in the definition are the roots of x^3-3*x^2+3. - Ralf Stephan, Apr 10 2004
a(n) = 3*(a(n-1) - a(n-3)) for n >= 4 - Roman Witula, Sep 29 2012
Extensions
More terms from Sascha Kurz, Mar 24 2002
Adapted formula, denominator of g.f. and modified g.f. (and offset) to accommodate added initial term a(0)=4. - L. Edson Jeffery, Apr 05 2011
a(0) = 4 removed, g.f. and programs adapted by Georg Fischer, Feb 02 2019
Comments
References
Links
Crossrefs
Programs
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 *)PARI
Formula