A110048 Expansion of 1/((1+2*x)*(1-4*x-4*x^2)).
1, 2, 16, 64, 336, 1568, 7680, 36864, 178432, 860672, 4157440, 20070400, 96915456, 467935232, 2259419136, 10909384704, 52675280896, 254338531328, 1228055511040, 5929575645184, 28630525673472, 138240403177472
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Robert Munafo, Sequences Related to Floretions
- Index entries for linear recurrences with constant coefficients, signature (2,12,8).
Programs
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Magma
[2^(n-2)*(Evaluate(DicksonFirst(n+1,-1), 2) +2*(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 18 2022
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Maple
seriestolist(series(1/((1+2*x)*(1-4*x-4*x^2)), x=0,40));
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Mathematica
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 *) -
SageMath
[2^(n-2)*(lucas_number2(n+1,2,-1) +2*(-1)^n) for n in (0..40)] # G. C. Greubel, Aug 18 2022
Formula
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.)
From Maksym Voznyy (voznyy(AT)mail.ru), Jul 24 2008: (Start)
a(n) = 2*a(n-1) + 12*a(n-2) + 8*a(n-3), where a(1)=1, a(2)=2, a(3)=16.
a(n) = 2^(n-3)*( 4*(-1)^(1-n) + (sqrt(2)-1)^(-n) + (-sqrt(2)-1)^(-n)) . (End)
a(n) = 2^n*A097076(n+1). - R. J. Mathar, Mar 08 2021
Comments