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%I #22 Mar 11 2024 20:53:27
%S 1,2,16,64,336,1568,7680,36864,178432,860672,4157440,20070400,
%T 96915456,467935232,2259419136,10909384704,52675280896,254338531328,
%U 1228055511040,5929575645184,28630525673472,138240403177472
%N Expansion of 1/((1+2*x)*(1-4*x-4*x^2)).
%C Floretion Algebra Multiplication Program, FAMP Code:
%C -kbasejseq[A*B] with A = + 'i - .5'j + .5'k - .5j' + .5k' - 'ii' - .5'ij' - .5'ik' - .5'ji' - .5'ki' and B = - .5'i + .5'j + 'k - .5i' + .5j' - 'kk' - .5'ik' - .5'jk' - .5'ki' - .5'kj'
%C See also comment for A110047.
%H G. C. Greubel, <a href="/A110048/b110048.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 (2,12,8).
%F Superseeker finds: a(n+1) = 2*A086348(n+1) (A086348's offset is 1: On a 3 X 3 board, number of n-move routes of chess king ending at central cell); binomial transform matches A084159 (Pell oblongs); j-th coefficient of g.f.*(1+x)^j matches A079291 (Squares of Pell numbers); a(n) + a(n+1) = A086346(n+2) (A086346's offset is 1: On a 3 X 3 board, the number of n-move paths for a chess king ending in a given corner cell.)
%F From Maksym Voznyy (voznyy(AT)mail.ru), Jul 24 2008: (Start)
%F a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3), where a(1)=1, a(2)=2, a(3)=16.
%F a(n) = 2^(n-3)*( 4*(-1)^(1-n) + (sqrt(2)-1)^(-n) + (-sqrt(2)-1)^(-n)) . (End)
%F a(n) = 2^n*A097076(n+1). - _R. J. Mathar_, Mar 08 2021
%p seriestolist(series(1/((1+2*x)*(1-4*x-4*x^2)), x=0,40));
%t CoefficientList[Series[1/((1+2x)(1-4x-4x^2)), {x,0,40}], x] (* or *) LinearRecurrence[{2,12,8}, {1,2,16}, 41] (* _Harvey P. Dale_, Nov 02 2011 *)
%o (Magma) [2^(n-2)*(Evaluate(DicksonFirst(n+1,-1), 2) +2*(-1)^n): n in [0..40]]; // _G. C. Greubel_, Aug 18 2022
%o (SageMath) [2^(n-2)*(lucas_number2(n+1,2,-1) +2*(-1)^n) for n in (0..40)] # _G. C. Greubel_, Aug 18 2022
%Y Cf. A079291, A084159, A086346, A086348, A110046, A110047, A110049, A110050.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Jul 10 2005
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