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A217787 a(n) = (a(n-1)*a(n-3) + 1) / a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.

%I A217787 #45 Sep 08 2022 08:46:04
%S A217787 1,1,1,1,2,3,4,9,14,19,43,67,91,206,321,436,987,1538,2089,4729,7369,
%T A217787 10009,22658,35307,47956,108561,169166,229771,520147,810523,1100899,
%U A217787 2492174,3883449,5274724,11940723,18606722,25272721,57211441,89150161,121088881
%N A217787 a(n) = (a(n-1)*a(n-3) + 1) / a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.
%C A217787 This sequence is similar to A005246 whose recursion is a(n) = (a(n-1)*a(n-2) + 1) / a(n-3). - _Michael Somos_, Feb 10 2017
%H A217787 Seiichi Manyama, <a href="/A217787/b217787.txt">Table of n, a(n) for n = 0..4411</a>
%H A217787 Matthew Christopher Russell, <a href="https://pdfs.semanticscholar.org/fdeb/e20954dacb7ec7a24afe2cf491b951c5a28d.pdf">Using experimental mathematics to conjecture and prove theorems in the theory of partitions and commutative and non-commutative recurrences</a>, PhD Dissertation, Mathematics Department, Rutgers University, May 2016. See (better) <a href="http://www.math.rutgers.edu/~zeilberg/Theses/MatthewRussellThesis.pdf">also</a>. See Example 6.2.18.
%H A217787 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,5,0,0,-1).
%F A217787 G.f.: (1 + x + x^2 - 4*x^3 - 3*x^4 - 2*x^5) / (1 - 5*x^3 + x^6).
%F A217787 a(n) = a(3-n) for all n in Z.
%F A217787 a(n+3) + a(n-3) = 5*a(n) for all n in Z.
%F A217787 a(n+1) + a(n-1) = a(n) * (2 + [n mod 3 == 0]) for all n in Z.
%F A217787 a(n+3k)+a(n-3k) = A003501(k)*a(n) for n>=3k. - _Bruno Berselli_, Mar 25 2013
%e A217787 G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 9*x^7 + 14*x^8 + 19*x^9 + ...
%t A217787 a[ n_] := With[{m = If [n < 0, 3 - n, n]}, SeriesCoefficient[ (1 + x + x^2 - 4 x^3 - 3 x^4 - 2 x^5) / (1 - 5 x^3 + x^6), {x, 0, m}]]; (* _Michael Somos_, Jan 18 2015 *)
%t A217787 LinearRecurrence[{0,0,5,0,0,-1},{1,1,1,1,2,3},40] (* _Harvey P. Dale_, Nov 20 2016 *)
%o A217787 (PARI) {a(n) = if( n<0, n = 3-n); polcoeff( (1 + x + x^2 - 4*x^3 - 3*x^4 - 2*x^5) / (1 - 5*x^3 + x^6) + x * O(x^n), n)};
%o A217787 (Magma) [n le 3 select 1 else (Self(n)*Self(n-2)+1)/Self(n-3): n in [0..40]]; // _Bruno Berselli_, Mar 25 2013
%Y A217787 Cf. A003501.
%Y A217787 Cf. A005246, A048736, A006720, A072876, A072877, A111459.
%K A217787 nonn,easy
%O A217787 0,5
%A A217787 _Michael Somos_, Mar 25 2013