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%I #15 Sep 08 2022 08:46:16
%S 1,3,6,14,32,74,171,396,917,2124,4920,11397,26401,61158,141673,328187,
%T 760249,1761126,4079670,9450606,21892446,50714123,117479896,272143639,
%U 630424122,1460385314,3383000731,7836763241,18153959452,42053872709,97418318825,225670746387,522769088906,1211001092038
%N Expansion of (1 + x - x^2 - x^3 - x^4)/((1 - x)*(1 - x - 2*x^2 - 2*x^3 - x^4)).
%C Partial sums of A272642. - _Wolfdieter Lang_, May 06 2016
%H Vincenzo Librandi, <a href="/A272362/b272362.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,0,-1,-1).
%F G.f.: (1 + x - x^2 - x^3 - x^4)/((1 - x)*(1 - x - 2*x^2 - 2*x^3 - x^4)).
%F a(n) = 2*a(n-1) + a(n-2) - a(n-4) - a(n-5).
%F a(n) = floor(phi*a(n-1) + phi*a(n-2)), a(0)=1, a(1)=3, where phi is the golden ratio (A001622).
%F Lim_{n->infinity} a(n)/a(n-1) = 2/(sqrt(2*sqrt(5)-1) - 1) = sqrt(phi + phi*sqrt(phi + phi*sqrt(phi + ...))) = A189970.
%F Lim_{n->infinity} a(n-1)/a(n) = (sqrt(2*sqrt(5)-1) - 1)/2 = 1 + A190157.
%t LinearRecurrence[{2, 1, 0, -1, -1}, {1, 3, 6, 14, 32}, 34]
%t RecurrenceTable[{a[n] == Floor[GoldenRatio a[n - 1] + GoldenRatio a[n - 2]], a[0] == 1, a[1] == 3}, a, {n, 33}]
%t CoefficientList[Series[(1 + x - x^2 - x^3 - x^4)/((1 - x) (1 - x - 2 x^2 - 2 x^3 - x^4)), {x, 0, 50}], x] (* _Vincenzo Librandi_, May 08 2016 *)
%o (PARI) Vec((1+x-x^2-x^3-x^4)/(1-2*x-x^2+x^4+x^5) + O(x^99)) \\ _Altug Alkan_, Apr 27 2016
%o (Magma) I:=[1,3,6,14,32]; [n le 5 select I[n] else 2*Self(n-1)+Self(n-2)-Self(n-4)-Self(n-5): n in [1..30]]; // _Vincenzo Librandi_, May 08 2016 *)
%Y Cf. A000045, A000201, A001622, A189970, A190157, A272642.
%K nonn,easy
%O 0,2
%A _Ilya Gutkovskiy_, Apr 27 2016
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