A099016 - OEIS

%I #36 Feb 06 2022 06:50:52

%S 1,2,4,11,28,74,193,506,1324,3467,9076,23762,62209,162866,426388,

%T 1116299,2922508,7651226,20031169,52442282,137295676,359444747,

%U 941038564,2463670946,6449974273,16886251874,44208781348,115740092171

%N a(n) = 3*L(2*n)/5 - (-1)^n/5, where L = A000032.

%C Let M = an infinite triangle with (1,2,2,3,3,4,4,...) as the left border and all other columns = (0,1,2,3,4,5,...). Then lim_{n->infinity} M^n = A099016, the left-shifted vector considered as a sequence. - _Gary W. Adamson_, Jul 26 2010

%H G. C. Greubel, <a href="/A099016/b099016.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..300 from Vincenzo Librandi)

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

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

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

%F a(n) = 2*F(n)^2 + F(n)*F(n-1) + F(n-1)^2, where F = A000045.

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

%F a(n) = A099015(n)/F(n+1).

%F a(n) = 3*A005248(n)/5 - (-1)^n/5.

%F a(n) = 3*A000032(2*n)/5 - (-1)^n/5.

%F a(n) = A061646(n) + F(n)^2.

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

%F a(n) = F(n+1)^2 + F(n)*F(n-2). See also A192914, fourth formula. - _Bruno Berselli_, Feb 15 2017

%p with(combinat):seq(3*fibonacci(n)^2+(-1)^n, n= 0..27)

%t CoefficientList[Series[(1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2)), {x,0,50}], x] (* _G. C. Greubel_, Dec 31 2017 *)

%o (Magma) [3*Lucas(2*n)/5-(-1)^n/5: n in [0..35]]; // _Vincenzo Librandi_, Jun 09 2011

%o (Magma) F:=Fibonacci; [F(n+1)^2+F(n)*F(n-2): n in [0..30]]; // _Bruno Berselli_, Feb 15 2017

%o (PARI) x='x+O('x^30); Vec((1 - 2*x^2)/((1 + x)*(1 - 3*x + x^2))) \\ _G. C. Greubel_, Dec 31 2017

%Y Cf. A000032.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Sep 22 2004