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

%I A105635 #44 Feb 16 2025 08:32:57
%S A105635 0,1,2,6,14,35,84,204,492,1189,2870,6930,16730,40391,97512,235416,
%T A105635 568344,1372105,3312554,7997214,19306982,46611179,112529340,271669860,
%U A105635 655869060,1583407981,3822685022,9228778026,22280241074,53789260175,129858761424,313506783024
%N A105635 a(n) = (2*Pell(n+1) - (1+(-1)^n))/4.
%C A105635 Transform of Pell(n) under the Riordan array (1/(1-x^2), x).
%C A105635 Starting (1, 2, 6, 14, 35, ...) equals row sums of triangle A157901. - _Gary W. Adamson_, Mar 08 2009
%C A105635 Starting with 1 = row sums of a triangle with the Pell series shifted down twice for columns > 1. - _Gary W. Adamson_, Mar 03 2010
%C A105635 Also the matching and vertex cover numbers of the n-Pell graph. - _Eric W. Weisstein_, Aug 01 2023
%H A105635 G. C. Greubel, <a href="/A105635/b105635.txt">Table of n, a(n) for n = 0..1000</a>
%H A105635 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MatchingNumber.html">Matching Number</a>
%H A105635 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PellGraph.html">Pell Graph</a>
%H A105635 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/VertexCoverNumber.html">Vertex Cover Number</a>
%H A105635 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F A105635 G.f.: x/((1-x^2)*(1-2*x-x^2)).
%F A105635 a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4).
%F A105635 a(n) = Sum_{k=0..floor((n-1)/2)} Pell(n-2k).
%F A105635 a(n) = Sum_{k=0..n} Pell(k)*(1-(-1)^(n+k-1))/2.
%F A105635 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 A105635 a(n) = (A033539(n+3) - A097076(n+3))/2. - _Gary Detlefs_, Dec 19 2010
%F A105635 a(n) = floor(Pell(n)/2). - _Eric W. Weisstein_, Aug 01 2023
%p A105635 with(combinat): seq(iquo(fibonacci(n+1,2),2),n=0..30); # _Zerinvary Lajos_, Apr 20 2008
%p A105635 # second Maple program:
%p A105635 a:= n-> (<<1|1|0|0>, <3|0|1|0>, <1|0|0|0>, <1|0|0|1>>^n)[4, 1]:
%p A105635 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jul 24 2008
%t A105635 Table[(Fibonacci[n + 1, 2] - Fibonacci[n + 1, 0])/2, {n, 0, 30}] (* _G. C. Greubel_, Oct 27 2019 *)
%t A105635 Floor[Fibonacci[Range[20], 2]/2] (* _Eric W. Weisstein_, Aug 01 2023 *)
%t A105635 Table[(2 Fibonacci[n + 1, 2] - (-1)^n - 1)/4, {n, 0, 10}]  (* _Eric W. Weisstein_, Aug 01 2023 *)
%t A105635 CoefficientList[Series[x/(1 - 2 x - 2 x^2 + 2 x^3 + x^4), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 01 2023 *)
%t A105635 LinearRecurrence[{2, 2, -2, -1}, {0, 1, 2, 6, 14}, 20] (* _Eric W. Weisstein_, Aug 01 2023 *)
%o A105635 (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 A105635 (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 A105635 (Sage)
%o A105635 def A105635_list(prec):
%o A105635     P.<x> = PowerSeriesRing(ZZ, prec)
%o A105635     return P(x/((1-x^2)*(1-2*x-x^2))).list()
%o A105635 A105635_list(30) # _G. C. Greubel_, Oct 27 2019
%o A105635 (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 A105635 Cf. A000129.
%Y A105635 Cf. A157901. - _Gary W. Adamson_, Mar 08 2009
%K A105635 easy,nonn
%O A105635 0,3
%A A105635 _Paul Barry_, Apr 16 2005