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A105635 a(n) = (2*Pell(n+2) - (1+(-1)^n))/4. 4

%I

%S 0,1,2,6,14,35,84,204,492,1189,2870,6930,16730,40391,97512,235416,

%T 568344,1372105,3312554,7997214,19306982,46611179,112529340,271669860,

%U 655869060,1583407981,3822685022,9228778026,22280241074,53789260175

%N a(n) = (2*Pell(n+2) - (1+(-1)^n))/4.

%C Transform of Pell(n) under the Riordan array (1/(1-x^2), x).

%C Starting (1, 2, 6, 14, 35, ...) equals row sums of triangle A157901. - _Gary W. Adamson_, Mar 08 2009

%C Starting with 1 = row sums of a triangle with the Pell series shifted down twice for columns > 1. - _Gary W. Adamson_, Mar 03 2010

%H G. C. Greubel, <a href="/A105635/b105635.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).

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

%F a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4).

%F a(n) = Sum_{k=0..floor((n-1)/2)} Pell(n-2k).

%F a(n) = Sum_{k=0..n} Pell(k)*(1-(-1)^(n+k-1))/2.

%F a(n) = term (4,1) in the 4 X 4 matrix [1,1,0,0; 3,0,1,0; 1,0,0,0; 1,0,0,1]^n. - _Alois P. Heinz_, Jul 24 2008

%F a(n) = ( A033539(n+3) - A09776(n+3) )/2. - _Gary Detlefs_ Dec 19 2010

%p with(combinat): seq(iquo(fibonacci(n+1,2),2),n=0..30); # _Zerinvary Lajos_, Apr 20 2008

%p # second Maple program:

%p a:= n-> (Matrix([[1,1,0,0], [3,0,1,0], [1,0,0,0], [1,0,0,1]])^(n))[4,1]; seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 24 2008

%t Table[(Fibonacci[n+1,2] - Fibonacci[n+1,0])/2, {n, 0,30}] (* _G. C. Greubel_, Oct 27 2019 *)

%o (PARI) my(x='x+O('x^30)); concat([0], Vec(x/((1-x^2)*(1-2*x-x^2)))) \\ _G. C. Greubel_, Oct 27 2019

%o (MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); [0] cat Coefficients(R!( x/((1-x^2)*(1-2*x-x^2)) )); // _G. C. Greubel_, Oct 27 2019

%o (Sage)

%o def A105635_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P(x/((1-x^2)*(1-2*x-x^2))).list()

%o A105635_list(30) # _G. C. Greubel_, Oct 27 2019

%o (GAP) a:=[0,1,2,6];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-2*a[n-3] -a[n-4]; od; a; # _G. C. Greubel_, Oct 27 2019

%Y Cf. A000129.

%Y Cf. A157901. - _Gary W. Adamson_, Mar 08 2009

%K easy,nonn

%O 0,3

%A _Paul Barry_, Apr 16 2005

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Last modified March 4 07:25 EST 2021. Contains 341781 sequences. (Running on oeis4.)