|
%I #25 Jan 31 2023 06:00:16
%S 0,1,2,20,72,464,2080,11584,56448,298240,1499648,7771136,39536640,
%T 203411456,1039409152,5333401600,27297349632,139929124864,
%U 716615843840,3672097685504,18810048872448,96373660712960,493708103385088,2529394778177536,12958119210516480
%N a(n+1) = 2*a(n) + 16*a(n-1), a(0)=0, a(1)=1.
%H Colin Barker, <a href="/A161007/b161007.txt">Table of n, a(n) for n = 0..1000</a>
%H J. Borowska, L. Lacinska, <a href="https://doi.org/10.17512/jamcm.2014.4.03">Recurrence form of determinant of a heptadiagonal symmetric Toeplitz matrix</a>, J. Appl. Math. Comp. Mech. 13 (2014) 19-16, remark 2 for permanent of tridiagonal Toeplitz matrices a=2, b=4.
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,16).
%F a(n) = ((1+sqrt(17))^n - (1-sqrt(17))^n)/(2*sqrt(17)).
%F Limit_{n -> oo} a(n+1)/a(n) = 1 + sqrt(17).
%F G.f.: x / (1 - 2*x - 16*x^2). - _Colin Barker_, Jul 01 2015
%F a(n) = 2^(n-1)*A006131(n-1). - _R. J. Mathar_, Mar 08 2021
%F a(n) = (4*i)^n*ChebyshevU(n, -i/4). - _G. C. Greubel_, Oct 15 2022
%t LinearRecurrence[{2, 16}, {0, 1}, 50] (* _T. D. Noe_, Nov 07 2011 *)
%o (PARI) concat(0, Vec(-x/(16*x^2+2*x-1) + O(x^40))) \\ _Colin Barker_, Jul 01 2015
%o (Magma) [n le 2 select n-1 else 2*(Self(n-1) +8*Self(n-2)): n in [1..41]]; // _G. C. Greubel_, Oct 15 2022
%o (SageMath)
%o A161007=BinaryRecurrenceSequence(2,16,0,1)
%o [A161007(n) for n in range(41)] # _G. C. Greubel_, Oct 15 2022
%Y Cf. A006131, A010473 (sqrt(17)).
%K nonn,easy
%O 0,3
%A _Sture Sjöstedt_, Jun 02 2009
|
