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A180678 The Ze2 sums of the Pell-Jacobsthal triangle A013609.

%I A180678 #7 Sep 08 2022 08:45:54
%S A180678 1,2,5,16,57,206,737,2612,9213,32442,114205,402072,1415713,4985126,
%T A180678 17554489,61816252,217679141,766531986,2699251381,9505089568,
%U A180678 33471028105,117864194430,415044573969,1461529529924,5146600421325
%N A180678 The Ze2 sums of the Pell-Jacobsthal triangle A013609.
%C A180678 The a(n) represent the Ze2 sums of the Pell-Jacobsthal triangle A013609. See A180662 for information about these zebra and other chess sums.
%H A180678 G. C. Greubel, <a href="/A180678/b180678.txt">Table of n, a(n) for n = 0..1000</a>
%H A180678 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,8).
%F A180678 a(n) = 6*a(n-1) - 11*a(n-2) + 8*a(n-3) with a(0)=1, a(1)=2 and a(2)= 5.
%F A180678 a(n) = Sum_{k=0..floor(n/2)} A013609(n+k,n-2*k).
%F A180678 G.f.: (1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3).
%p A180678 nmax:=24: a(0):=1: a(1):=2: a(2):=5: for n from 3 to nmax do a(n) := 6*a(n-1)-11*a(n-2)+8*a(n-3) od: seq(a(n),n=0..nmax);
%t A180678 LinearRecurrence[{6,-11,8}, {1,2,5}, 30] (* or *) CoefficientList[ Series[(1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3), {x,0,30}], x] (* _G. C. Greubel_, Jun 06 2019 *)
%o A180678 (PARI) my(x='x+O('x^30)); Vec((1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3)) \\ _G. C. Greubel_, Jun 06 2019
%o A180678 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3) )); // _G. C. Greubel_, Jun 06 2019
%o A180678 (Sage) ((1-4*x+4*x^2)/(1-6*x+11*x^2-8*x^3)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Jun 06 2019
%o A180678 (GAP) a:=[1,2,5];; for n in [4..30] do a[n]:=6*a[n-1]-11*a[n-2]+8*a[n-3]; od; a; # _G. C. Greubel_, Jun 06 2019
%Y A180678 Cf. A140413 (Ze1), A180678 (Ze2), A097117 (Ze3), A055588 (Ze4).
%K A180678 easy,nonn
%O A180678 0,2
%A A180678 _Johannes W. Meijer_, Sep 21 2010