A136161 - OEIS

%I #30 Dec 31 2023 10:22:32

%S 0,5,2,1,3,1,2,1,0,3,-1,-1,4,-3,-2,5,-5,-3,6,-7,-4,7,-9,-5,8,-11,-6,9,

%T -13,-7,10,-15,-8,11,-17,-9,12,-19,-10,13,-21,-11,14,-23,-12,15,-25,

%U -13,16,-27,-14

%N a(n) = 2*a(n-3) - a(n-6), starting a(0..5) = 0, 5, 2, 1, 3, 1.

%C Consider the general recurrence a(n) = k*a(n-1) + (5-2*k)*a(n-2) + (2-k)*a(n-3). The coefficients, in k, can be used to form the triple (k, 5-2*k, 2-k). Each triple is associated with a sequence, for example (0, 5, 2) leads to A111108, A112685, ..., (1, 3, 1) leads to A051927, A097075, ..., and so on. This sequence is formed from the triples {(0, 5, 2), (1, 3, 1), (2, 1, 0), (3, -1, -1), (4, -3, -2), ...}, for k >= 0. (Comment modified by _G. C. Greubel_, Dec 31 2023).

%H G. C. Greubel, <a href="/A136161/b136161.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,-1).

%F G.f.: x*(5+2*x+x^2-7*x^3-3*x^4) / ( (1-x)^2*(1+x+x^2)^2 ). - _R. J. Mathar_, Jul 06 2011

%F a(3n) = n.

%F a(3n+1) = 5 - 2*n.

%F a(3n+3) = 2 - n.

%F a(n) = (1/9)*( 27 - 2*(n+1) - 34*ChebyshevU(n, -1/2) + (-1)^n*(9*A099254(n) - 6*A099254(n-1)) ). - _G. C. Greubel_, Dec 26 2023

%t LinearRecurrence[{0,0,2,0,0,-1},{0,5,2,1,3,1},60] (* _Harvey P. Dale_, Aug 16 2012 *)

%t Table[PadRight[{n, 5-2*n, 2-n}], {n,0,20}]//Flatten (* _G. C. Greubel, Dec 26 2023 *)

%o (PARI) Vec(x*(5+2*x+x^2-7*x^3-3*x^4)/((1-x)^2*(1+x+x^2)^2+O(x^99))) \\ _Charles R Greathouse IV_, Jul 06 2011

%o (Magma) I:=[0,5,2,1,3,1]; [n le 6 select I[n] else 2*Self(n-3) - Self(n-6): n in [1..60]]; // _G. C. Greubel_, Dec 26 2023

%o (SageMath)

%o def a(n): # a = A136161

%o     if n<6: return (0,5,2,1,3,1)[n]

%o     else: return 2*a(n-3) - a(n-6)

%o [a(n) for n in range(61)] # _G. C. Greubel_, Dec 26 2023

%Y Cf. A097076, A099254, A112685, A135138, A135997, A137241.

%Y Cf. A051927, A097075, A111108, A112685.

%K sign,easy

%O 0,2

%A _Paul Curtz_, Mar 16 2008