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%I #18 Mar 14 2024 13:53:10
%S 1,4,7,18,41,100,239,578,1393,3364,8119,19602,47321,114244,275807,
%T 665858,1607521,3880900,9369319,22619538,54608393,131836324,318281039,
%U 768398402,1855077841,4478554084,10812186007,26102926098,63018038201
%N Expansion of (1+2*x-2*x^3-3*x^2)/((x-1)*(x+1)*(x^2+2*x-1)).
%C A floretion-generated sequence relating NSW and Pell numbers.
%C Elements of odd index in the sequence gives A002315. a(n+2) - a(n) = A002203(n+2).
%C Floretion Algebra Multiplication Program, FAMP Code: 2tesseq[B*C} with B = - .25'i + .25'j + .5'k - .25i' + .25j' + .5k' - .5'kk' - .25'ik' - .25'jk' - .25'ki' - .25'kj' - .5e and C = + .5'i - .25'j + .25'k + .5i' - .25j' + .25k' - .5'ii' - .25'ij' - .25'ik' - .25'ji' - .25'ki' - .5e
%H Colin Barker, <a href="/A100828/b100828.txt">Table of n, a(n) for n = 0..1000</a>
%H Robert Munafo, <a href="http://www.mrob.com/pub/math/seq-floretion.html">Sequences Related to Floretions</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-2,-1).
%F a(n) = (u^(n+1)+1)*(v^(n+1)+1)/2 with u = 1+sqrt(2), v = 1-sqrt(2). - _Vladeta Jovovic_, May 30 2007
%F From _Colin Barker_, Apr 29 2019: (Start)
%F G.f.: (1 + 2*x - 3*x^2 - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x - x^2)).
%F a(n) = (1 + (-1)^(1+n) + (1-sqrt(2))^(1+n) + (1+sqrt(2))^(1+n)) / 2.
%F a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) for n>3.
%F (End)
%o (PARI) Vec((1 + 2*x - 3*x^2 - 2*x^3) / ((1 - x)*(1 + x)*(1 - 2*x - x^2)) + O(x^30)) \\ _Colin Barker_, Apr 29 2019
%Y Cf. A002315, A002203.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Jan 06 2005; revised Aug 22 2005