A189234 Expansion of (5-4*x-12*x^2+6*x^3+3*x^4)/(1-x-4*x^2+3*x^3+3*x^4-x^5).
5, 1, 9, 4, 25, 16, 78, 64, 257, 256, 874, 1013, 3034, 3953, 10684, 15229, 38017, 58056, 136338, 219508, 491870, 824737, 1782735, 3083887, 6484514, 11489516, 23652443, 42688039, 86459608, 158270401, 316576903, 585868009, 1160673633, 2166063365, 4259693562
Offset: 0
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, 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
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
- Index entries for linear recurrences with constant coefficients, signature (4, -2, -5, 2, 1).
Programs
-
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 *) LinearRecurrence[{4,-2,-5,2,1},{4,12,25,64,159},30] (* Harvey P. Dale, Dec 30 2024 *) -
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
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
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
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
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
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
- L. E. Jeffery, Unit-primitive matrices
- Index entries for linear recurrences with constant coefficients, signature (4, -2, -5, 2, 1).
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
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
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).
A189237 Expansion of (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5).
5, 3, 15, 42, 155, 533, 1884, 6604, 23219, 81555, 286555, 1006734, 3537032, 12426742, 43659386, 153390077, 538911123, 1893376346, 6652069455, 23370962220, 82110068595, 288480349402, 1013528712002, 3560868017067, 12510529683224
Offset: 0
Comments
(Start) Let U be the unit-primitive matrix (see [Jeffery])
U=U_(11,4)=
(0 0 0 0 1)
(0 0 0 1 1)
(0 0 1 1 1)
(0 1 1 1 1)
(1 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 (3, 3, -4, -1, 1).
Programs
-
Mathematica
CoefficientList[Series[(5-12x-9x^2+8x^3+x^4)/(1-3x-3x^2+4x^3+x^4-x^5), {x,0,30}],x] (* or *) LinearRecurrence[{3,3,-4,-1,1},{5,3,15,42,155},30] (* Harvey P. Dale, Oct 01 2011 *) -
PARI
Vec((5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5)+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
Formula
G.f.: (5-12*x-9*x^2+8*x^3+x^4)/(1-3*x-3*x^2+4*x^3+x^4-x^5).
a(n)=3*a(n-1)+3*a(n-2)-4*a(n-3)-a(n-4)+a(n-5), {a(m)}={5,3,15,42,155}, m=0..4.
a(n)=Sum_{k=1..5} ((x_k)^4-3*(x_k)^2+1)^n; x_k=2*(-1)^(k-1)*cos(k*Pi/11).
Series expansion of g.f. at x=infinity gives -A189234(n+1).
Comments
Links
Crossrefs
Programs
Mathematica
CoefficientList[Series[(5-4x-12x^2+6x^3+3x^4)/(1-x-4x^2+3x^3+ 3x^4-x^5),{x,0,40}],x] (* or *) LinearRecurrence[{1,4,-3,-3,1},{5,1,9,4,25},40] (* Harvey P. Dale, Jan 18 2012 *)PARI
Formula