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 A110048 Expansion of 1/((2*x+1)*(1-4*x-4*x^2)). 3

%I

%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/((2*x+1)*(1-4*x-4*x^2)).

%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 Recurrence: a(n)=2*a(n-1)+12*a(n-2)+8*a(n-3), where a(1)=1, a(2)=2, a(3)=16; formula a(n)=(1/4)*(-1)^(1-n)*2^n+(1/8)*2^n*(sqrt(2)-1)^(-n)+(1/8)*2^n*(-sqrt(2)-1)^(-n). - Maksym Voznyy (voznyy(AT)mail.ru), Jul 24 2008

%p seriestolist(series(-1/((2*x+1)*(4*x^2+4*x-1)), x=0,25)); -or- Floretion Algebra Multiplication Program, FAMP Code: -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'

%t CoefficientList[Series[1/((2x+1)(1-4x-4x^2)),{x,0,30}],x] (* or *) LinearRecurrence[ {2,12,8},{1,2,16},30] (* _Harvey P. Dale_, Nov 02 2011 *)

%Y Cf. A084159, A086346, A079291, A110046, A110047, A110049, A110050, A086348.

%K easy,nonn

%O 0,2

%A _Creighton Dement_, Jul 10 2005

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Last modified August 10 16:56 EDT 2020. Contains 336381 sequences. (Running on oeis4.)