A189316 - OEIS

%I #33 Oct 20 2024 01:06:15

%S 5,5,15,35,95,245,645,1685,4415,11555,30255,79205,207365,542885,

%T 1421295,3720995,9741695,25504085,66770565,174807605,457652255,

%U 1198149155,3136795215,8212236485,21499914245,56287506245,147362604495,385800307235,1010038317215

%N Expansion of g.f. 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).

%C (Start) Let A be the unit-primitive matrix (see [Jeffery])

%C A=A_(10,2)=

%C (0 0 1 0 0)

%C (0 1 0 1 0)

%C (1 0 1 0 1)

%C (0 1 0 2 0)

%C (0 0 2 0 1).

%C Then a(n)=Trace(A^n). For m=1,2,..., A^(m) can also be written

%C A^(m)=

%C [   F(m-1)^2     0     F(m)^2      0      F(m-1)*F(m)      ]

%C [       0     F(2*m-1)    0      F(2*m)        0           ]

%C [    F(m)^2      0    F(m+1)^2     0      F(m)*F(m+1)      ]

%C [       0      F(2*m)     0     F(2*m+1)       0           ]

%C [ 2*F(m-1)*F(m)  0  2*F(m)*F(m+1)  0  F(2*m+1)-F(m)*F(m+1) ],

%C where F(m-1)=A000045(n) are the Fibonacci numbers and m=n+1. Hence also a(n+1)=Trace(A^(n+1))=F(m-1)^2+F(2*m-1)+F(m+1)^2+2*F(2*m+1)-F(m)*F(m+1). (End)

%C Evidently one of a class of accelerator sequences for Catalan's constant based on traces of successive powers of a unit-primitive matrix A_(N,r), 0<r<Floor(N/2), and for which the closed-form expression for a(n) is derived from the eigenvalues of A_(N,r).

%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-1).

%F G.f.: 5*(1-x-x^2)/((1+x)*(1-3*x+x^2)).

%F a(n) = 2*a(n-1)+2*a(n-2)-a(n-3), n>2, a(0)=5, a(1)=5, a(2)=15.

%F a(n) = Sum_{k=1..5} ((w_k)^2-1)^n, w_k = 2*cos((2*k-1)*Pi/10).

%F a(n) = (-1)^n+2*(1/tau^(2*n)+tau^(2*n)), tau = (1+sqrt(5))/2=1.618033....

%F a(n) = 5*A061646(n), n>=0 (offset for A061646 is -1).

%F E.g.f.: cosh(x) + 4*exp(3*x/2)*cosh(sqrt(5)*x/2) - sinh(x). - _Stefano Spezia_, Jul 09 2024

%t CoefficientList[Series[5 (1-x-x^2)/((1+x)(1-3x+x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{2,2,-1},{5,5,15},40] (* _Harvey P. Dale_, Nov 26 2016 *)

%Y Cf. A000045, A061646, A189315, A189317, A189318.

%K nonn,easy

%O 0,1

%A _L. Edson Jeffery_, Apr 20 2011