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A100213 Expansion of g.f.: x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-2*x) * (1-2*x^2) * (1-2*x+2*x^2) * (1+2*x^2)).

%I A100213 #22 Mar 30 2024 05:32:20
%S A100213 4,9,14,18,32,64,128,256,544,1104,2144,4128,8192,16384,32768,65536,
%T A100213 131584,263424,525824,1049088,2097152,4194304,8388608,16777216,
%U A100213 33562624,67129344,134242304,268443648,536870912,1073741824,2147483648,4294967296,8590065664
%N A100213 Expansion of g.f.: x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-2*x) * (1-2*x^2) * (1-2*x+2*x^2) * (1+2*x^2)).
%C A100213 The sequence can be created applying the pos operator (which sums over the positive coefficients) to the n-th power of the Floretion element (.5 'j + .5 'k + .5 j' + .5 k' + 1 'ii' + 1 e).
%H A100213 G. C. Greubel, <a href="/A100213/b100213.txt">Table of n, a(n) for n = 1..1000</a>
%H A100213 Creighton Dement, <a href="https://web.archive.org/web/20201230160447/http://fumba.eu/sitelayout/Floretion.php">Floretion Online Multiplier</a>.
%H A100213 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,4,-16,24,-16).
%F A100213 a(n) = A100215(n) - A100212(n).
%F A100213 a(n) = (-1)^n*A009116(n+3) + A100216 + A038503(n+1).
%F A100213 Equation above in Floretian Algebra operator speak: (pos) + (neg) = (ves) = (jes) + (les) + (tes)
%F A100213 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
%F A100213 From _G. C. Greubel_, Mar 29 2024: (Start)
%F A100213 a(n) = (1/16)*( 2^(n+4) - 2*((1+5*i)*(1+i)^n + (1-5*i)*(1-i)^n) + (1 - (-1)^n)*2^((n+1)/2)*(5+i^(n+1)) + (1+(-1)^n)*2^(1+n/2)*(3-2*i^n) ).
%F A100213 a(2*n-1) = 2^(n-3)*( 2^(n+2) + 5 + (-1)^n - 6*cos(n*Pi/2) + 4*sin(n*Pi/2) ), for n >= 1.
%F A100213 a(2*n) = 2^(n-2)*( 2^(n+2) + 3 - 2*(-1)^n - cos(n*Pi/2) + 5*sin(n*Pi/2) ), n >= 1.
%F A100213 E.g.f.: -1 + exp(2*x) + (1/8)*(6*cosh(sqrt(2)*x) + 5*sqrt(2)* sinh(sqrt(2)*x) - (4*cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x)) - 2*exp(x)*(cos(x) - 5*sin(x)) ). (End)
%e A100213 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 A100213 and the sum of all positive coefficients is 6+10+10+6 = 32.
%t A100213 Rest[CoefficientList[Series[x(4-7x+2x^2-8x^4+16x^5-16x^6)/((1-2x)(1-2x^2)(1-2x+2x^2)(1+2x^2)),{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 *)
%o A100213 (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) )); // _G. C. Greubel_, Mar 29 2024
%o A100213 (Sage)
%o A100213 def A100213_list(prec):
%o A100213     P.<x> = PowerSeriesRing(ZZ, prec)
%o A100213     return P( x*(4-7*x+2*x^2-8*x^4+16*x^5-16*x^6)/((1-4*x+6*x^2-4*x^3)*(1-4*x^4)) ).list()
%o A100213 a=A100213_list(51); a[1:] # _G. C. Greubel_, Mar 29 2024
%Y A100213 Cf. A000079, A009116, A009545, A038503, A077957, A077966, A100212, A100215, A100216.
%K A100213 nonn,easy
%O A100213 1,1
%A A100213 _Creighton Dement_, Nov 11 2004
%E A100213 Replaced definition with generating function, changed offset to 1. - _R. J. Mathar_, Mar 12 2010