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%I #43 Aug 12 2023 00:59:18
%S 1,1,0,1,2,2,3,5,7,10,15,22,32,47,69,101,148,217,318,466,683,1001,
%T 1467,2150,3151,4618,6768,9919,14537,21305,31224,45761,67066,98290,
%U 144051,211117,309407,453458,664575,973982,1427440,2092015,3065997,4493437
%N Expansion of (1 - x^2)/(1 - x - x^3).
%H G. C. Greubel, <a href="/A058278/b058278.txt">Table of n, a(n) for n = 0..1000</a>
%H Engin Özkan, Bahar Kuloǧlu, and James Peters, <a href="https://hal.archives-ouvertes.fr/hal-03242990">k-Narayana sequence self-similarity</a>, hal-03242990 [math.CO], 2021. See p. 12.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1).
%F G.f.: (1 - x^2)/(1 - x - x^3).
%F a(n+3) = Sum_{k=0..n} binomial(n-k, floor(k/2)). - _Paul Barry_, Jul 06 2004
%F a(n) = a(n-3) + a(n-1). - _Graeme McRae_, Apr 26 2010
%F From _Wolfdieter Lang_, Apr 21 2015 : (Start)
%F a(n) = A097333(n-3), n >= 3.
%F a(n) = A000930(n) - A000930(n-2), n >= 2. (End)
%F a(n) = A003410(n-4) for n >= 5. - _Jianing Song_, Aug 11 2023
%t CoefficientList[Series[(1 - x^2)/(1 - x - x^3), {x, 0, 50}], x] (* _Vladimir Joseph Stephan Orlovsky_, Dec 28 2010 *)
%t LinearRecurrence[{1,0,1},{1,1,0},50] (* _Harvey P. Dale_, Apr 03 2015 *)
%o (PARI) Vec((1-x^2)/(1-x-x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%Y Essentially the same as A003410 and A097333.
%Y Cf. A000930.
%K nonn,easy
%O 0,5
%A _Robert G. Wilson v_, Dec 06 2000
%E Edited: Offset corrected to 0. The formula by P. Barry corrected. Old formulas adapted to new offset. - _Wolfdieter Lang_, Apr 21 2015
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