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A274203 Expansion of x*(1 - x - x^3)/((1 - x)*(1 - 2*x - 3*x^2 - 2*x^3 - x^4)). 0

%I

%S 0,1,2,7,21,67,212,673,2136,6781,21527,68341,216960,688777,2186642,

%T 6941875,22038189,69964063,222113084,705136609,2238578784,7106757625,

%U 22561637903,71625842857,227388693456,721884948913,2291749301810,7275556680127,23097519856965,73327093306843,232789608846644

%N Expansion of x*(1 - x - x^3)/((1 - x)*(1 - 2*x - 3*x^2 - 2*x^3 - x^4)).

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-1,-1,-1)

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

%F a(n) = 3*a(n-1) + a(n-2) - a(n-3) - a(n-4) - a(n-5).

%F a(n) = floor((1 + sqrt(2))*a(n-1) + (1 + sqrt(2))*a(n-2)), a(0)=0, a(1)=1 (empirically).

%F Lim_{n->infinity} a(n)/a(n+1) = sqrt(sqrt(2) - sqrt(sqrt(2) + sqrt(sqrt(2) - sqrt(sqrt(2) + ...)))) = (sqrt(4*sqrt(2) - 3) - 1)/2 = A190179 - 1.

%t LinearRecurrence[{3, 1, -1, -1, -1}, {0, 1, 2, 7, 21}, 31]

%t RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == Floor[(Sqrt[2] + 1) a[n - 1] + (Sqrt[2] + 1) a[n - 2]]}, a, {n, 30}]

%o (PARI) concat(0, Vec(x*(1-x-x^3)/((1-x)*(1-2*x-3*x^2-2*x^3-x^4)) + O(x^99))) \\ _Altug Alkan_, Jun 26 2016

%Y Cf. A000045, A003151, A014176, A190179, A272362.

%K nonn,easy

%O 0,3

%A _Ilya Gutkovskiy_, Jun 13 2016

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Last modified October 20 22:44 EDT 2019. Contains 328291 sequences. (Running on oeis4.)