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%I #18 Sep 08 2022 08:45:29
%S 1,2,2,4,5,9,13,22,34,56,89,145,233,378,610,988,1597,2585,4181,6766,
%T 10946,17712,28657,46369,75025,121394,196418,317812,514229,832041,
%U 1346269,2178310,3524578,5702888,9227465,14930353,24157817,39088170,63245986,102334156
%N a(n) = F(n+1) + (1-(-1)^n)/2, where F() = Fibonacci numbers A000045.
%C Row sums of A127967.
%C The sequence beginning 1,1,2,2,4,... with g.f. x/(1-x-x^2) + 1/(1-x^2) has general term a(n) = F(n) + (1+(-1)^n)/2.
%H Colin Barker, <a href="/A127968/b127968.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 (1,2,-1,-1).
%F G.f.: 1 / (1 - x - x^2) + x / (1 - x^2).
%F G.f.: (1 + x - 2*x^2 - x^3) / ((1 - x)*(1 + x)*(1 - x - x^2)).
%F From _Colin Barker_, Jul 12 2017: (Start)
%F a(n) = (5 - 5*(-1)^n + 2^(-n)*sqrt(5)*(-(1-sqrt(5))^(1+n) + (1+sqrt(5))^(1+n))) / 10.
%F a(n) = a(n-1) + 2*a(n-2) - a(n-3) - a(n-4) for n>3.
%F (End)
%t LinearRecurrence[{1,2,-1,-1},{1,2,2,4},40] (* _Harvey P. Dale_, Jun 19 2013 *)
%o (PARI) Vec((1+x-2*x^2-x^3)/((1-x)*(1+x)*(1-x-x^2)) + O(x^50)) \\ _Colin Barker_, Jul 12 2017
%o (Magma) I:=[1,2,2,4]; [n le 4 select I[n] else Self(n-1) +2*Self(n-2) - Self(n-3) -Self(n-4): n in [1..30]]; // _G. C. Greubel_, May 04 2018
%K easy,nonn
%O 0,2
%A _Paul Barry_, Feb 09 2007
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