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%I #25 Mar 15 2024 15:20:24
%S -1,1,-25,49,-1,529,-1849,289,-9025,58081,-38809,108241,-1560001,
%T 2283121,-525625,35796289,-95863681,2666689,-681575449,3261894769,
%U -1289169025,9906021841,-94109673529,99199171681,-84332740801,2327696411041,-4753075824025,46970592529,-48635546218561
%N Expansion of g.f. -(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2)).
%C Floretion Algebra Multiplication Program, FAMP Code: 1tesseq[A*B] with A = + .5'i - .5'k + .5i' - .5k' - 3'jj' - .5'ij' - .5'ji' - .5'jk' - .5'kj' and B = + .5'i + .5'j + .5i' + .5j' + .5'kk' + .5'ij' + .5'ji' + .5e
%D S. Severini, A note on two integer sequences arising from the 3-dimensional hypercube, Technical Report, Department of Computer Science, University of Bristol, Bristol, UK (October 2003).
%H G. C. Greubel, <a href="/A106632/b106632.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_03">Index entries for linear recurrences with constant coefficients</a>, signature (-1,-3,-27).
%F a(n) = (3^(n+1)/2)*(cos((n+1)*arccos(1/3)) + (-1)^(n+1) ).
%F a(n) = - a(n-1) - 3*a(n-2) - 27*a(n-3), a(0) = -1, a(1) = 1, a(2) = -25.
%F a(n) = 1/4( p^(n+1) + q^(n+1) ) + (-3)^(n+1)/2 with p = 1 + 2*sqrt(2)i and q = 1 - 2*sqrt(2)i ( i^2 = -1 ).
%F a(n) = ((-1)^(n+1))*(A087455(n+1))^2; 2*a(n) = A025172(n) + (-3)^(n+1).
%t CoefficientList[Series[-(1+27x^2)/((1+3x)(1-2x+9x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{-1,-3,-27},{-1,1,-25},40] (* _Harvey P. Dale_, Oct 03 2014 *)
%o (PARI) my(x='x+O('x^40)); Vec(-(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2))) \\ _G. C. Greubel_, Feb 19 2019
%o (Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( -(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2)) )); // _G. C. Greubel_, Feb 19 2019
%o (SageMath) (-(1+27*x^2)/((1+3*x)*(1-2*x+9*x^2))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 19 2019
%o (GAP) a:=[-1,1,-25];; for n in [4..40] do a[n]:=-a[n-1]-3*a[n-2] - 27*a[n-3]; od; a; # _G. C. Greubel_, Feb 19 2019
%Y Cf. A087455, A025172.
%K sign,easy
%O 0,3
%A _Creighton Dement_, May 11 2005
%E Edited by _Ralf Stephan_, Apr 09 2009
%E Definition corrected by _Harvey P. Dale_, Oct 03 2014
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