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A100213 G.f. x* (4-7*x+2*x^2-8*x^4+16*x^5-16*x^6) / ((2*x-1) * (2*x^2-1) * (2*x^2-2*x+1) * (2*x^2+1)). 3

%I

%S 4,9,14,18,32,64,128,256,544,1104,2144,4128,8192,16384,32768,65536,

%T 131584,263424,525824,1049088,2097152,4194304,8388608,16777216,

%U 33562624,67129344,134242304,268443648,536870912,1073741824,2147483648,4294967296,8590065664

%N G.f. x* (4-7*x+2*x^2-8*x^4+16*x^5-16*x^6) / ((2*x-1) * (2*x^2-1) * (2*x^2-2*x+1) * (2*x^2+1)).

%C The sequence can be created applying the pos operator (which sums over the positive coefficients)

%C to the n-th power of the Floretion element (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e).

%H Creighton Dement, <a href="http://fumba.eu/">Floretion Online Multiplier</a>.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4, -6, 4, 4, -16, 24, -16).

%F a(n) + A100212(n) = A100215(n) = ((-1)^n)*A009116(n+3) + A100216 + A038503(n+1).

%F Equation above in Floretian Algebra operator speak: (pos) + (neg) = (ves) = (jes) + (les) + (tes)

%F a(n-1) = A000079(n+1) + (5*A077957(n)+6*A077957(n-1))/4 + A009545(n)/2 + A009545(n+1) + A077966(n-1) - A077966(n)/4. - _R. J. Mathar_, May 07 2008

%e a(5) = 32 because (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e)^5 = - 2 'j - 2 'k - 2 j' - 2 k' + 6 'ii' + 10 'jj' + 10 'kk' + 6 e,

%e and the sum of all positive coefficients is 6+10+10+6 = 32.

%t Rest[CoefficientList[Series[x (4-7x+2x^2-8x^4+16x^5-16x^6)/((2x-1)(2x^2-1)(2x^2-2x+1)(2x^2+1)),{x,0,40}],x]] (* or *) LinearRecurrence[{4,-6,4,4,-16,24,-16},{4,9,14,18,32,64,128},40] (* _Harvey P. Dale_, Aug 23 2015 *)

%Y Cf. A100212, A100215, A100216, A009116, A038503.

%K nonn,easy

%O 1,1

%A _Creighton Dement_, Nov 11 2004

%E Replaced definition with generating function, changed offset to 1. - _R. J. Mathar_, Mar 12 2010

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Last modified September 19 15:08 EDT 2019. Contains 327198 sequences. (Running on oeis4.)