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

Author

Creighton Dement, Jul 10 2005

Keywords

• easy
• nonn

Comments

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'

See also comment for A110047.

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).

Crossrefs

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

Programs

Magma

[2^(n-2)*(Evaluate(DicksonFirst(n+1,-1), 2) +2*(-1)^n): n in [0..40]]; // G. C. Greubel, Aug 18 2022

Maple

seriestolist(series(1/((1+2*x)*(1-4*x-4*x^2)), x=0,40));

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