A137241 - OEIS

%I #15 Mar 03 2024 14:40:12

%S 0,3,2,1,2,0,2,1,-2,3,0,-4,4,-1,-6,5,-2,-8,6,-3,-10,7,-4,-12,8,-5,-14,

%T 9,-6,-16,10,-7,-18,11,-8,-20,12,-9,-22,13,-10,-24,14,-11,-26,15,-12,

%U -28,16,-13,-30,17,-14,-32,18,-15,-34,19,-16,-36,20,-17,-38,21,-18,-40

%N Number triples (k,3-k,2-2k), concatenated for k=0, 1, 2, 3,...

%C The entries are the coefficients in a family of Jacobsthal recurrences: a(n)=k*a(n-1)+(3-k)*a(n-2)+(2-2k)*a(n-3).

%C Examples for k=0 are in A001045 and A113954. Examples for k=1 are A001045, A078008.

%C Examples for k=2 are A000975, A087288, A084639, A000012 and A001045.

%C Examples for k=3 are A045883, A059570. Examples for k=4 are A094705 and A015518.

%H G. C. Greubel, <a href="/A137241/b137241.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 From _R. J. Mathar_, Feb 25 2009: (Start)

%F a(n) = 2*a(n-3) - a(n-6).

%F G.f.: x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x+x^2)^2). (End)

%e The triples (k,3-k,2-2k) are (0,3,2), (1,2,0), (2,1,-2), (3,0,-4),...

%t CoefficientList[Series[x*(3 + 2*x + x^2 - 4*x^3 - 4*x^4)/((x - 1)^2*(1 + x + x^2)^2), {x, 0, 50}], x] (* _G. C. Greubel_, Sep 28 2017 *)

%t Table[{n,3-n,2-2n},{n,0,30}]//Flatten (* or *) LinearRecurrence[ {0,0,2,0,0,-1},{0,3,2,1,2,0},100] (* _Harvey P. Dale_, Jun 23 2019 *)

%o (PARI) x='x+O('x^50); Vec(x*(3+2*x+x^2-4*x^3-4*x^4)/((x-1)^2*(1+x +x^2 )^2)) \\ _G. C. Greubel_, Sep 28 2017

%K easy,sign,less

%O 0,2

%A _Paul Curtz_, Mar 09 2008

%E Edited by _R. J. Mathar_, Jun 28 2008