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Expansion of x/(1 - x + x^5 - x^6).
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%I #33 May 11 2024 17:16:14

%S 0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,

%T 1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,

%U 0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0

%N Expansion of x/(1 - x + x^5 - x^6).

%C Decimal expansion of 10000/900009. - _Elmo R. Oliveira_, May 08 2024

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,-1,1).

%F G.f.: x/(1 - x + x^5 - x^6) = x/((1-x)*(1+x)*(1-x+x^2-x^3+x^4)).

%F a(n) = a(n-1) - a(n-5) + a(n-6).

%F a(n) = a(n-10).

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

%F a(n) = A112712(n) mod 2.

%t CoefficientList[Series[x/(1 - x + x^5 - x^6), {x, 0, 100}], x] (* or *) LinearRecurrence[{1, 0, 0, 0, -1, 1}, {0, 1, 1, 1, 1, 1}, 100] (* _Harvey P. Dale_, Feb 16 2014 *)

%Y Cf. A000035, A133872, A088911, A131078, A112712.

%K nonn,easy

%O 0,1

%A _Paul Barry_, Sep 15 2005

%E Incorrect g.f. removed by _Georg Fischer_, May 15 2019