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A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)).

%I A117575 #57 Apr 20 2023 02:10:59
%S A117575 1,1,-1,-2,2,4,-4,-8,8,16,-16,-32,32,64,-64,-128,128,256,-256,-512,
%T A117575 512,1024,-1024,-2048,2048,4096,-4096,-8192,8192,16384,-16384,-32768,
%U A117575 32768,65536,-65536,-131072,131072,262144,-262144,-524288,524288
%N A117575 Expansion of (1-x^3)/((1-x)*(1+2*x^2)).
%C A117575 Row sums of A116949.
%C A117575 From _Paul Curtz_, Oct 24 2012: (Start)
%C A117575 b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
%C A117575 Consider the autosequence (that is a sequence whose inverse binomial transform is equal to the signed sequence) of the first kind of the example. Its numerator is A046978(n), its denominator is b(n). The numerator of the first column is A075553(n).
%C A117575 The denominator corresponding to the 0's is a choice.
%C A117575 The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)
%H A117575 G. C. Greubel, <a href="/A117575/b117575.txt">Table of n, a(n) for n = 0..1000</a>
%H A117575 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-2).
%F A117575 a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
%F A117575 a(n) = (cos(Pi*n/2) + sin(Pi*n/2)) * (2^((n-1)/2)*(1-(-1)^n)/2 + 2^((n-2)/2)*(1+(-1)^n)/2 + 0^n/2).
%F A117575 a(n+1) = Sum_{k=0..n} A122016(n,k)*(-1)^k. - _Philippe Deléham_, Jan 31 2012
%F A117575 E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - _Stefano Spezia_, Feb 05 2023
%F A117575 a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - _G. C. Greubel_, Apr 19 2023
%e A117575    0/1,  1/1    1/1,   1/2,   0/2,  -1/4,  -1/4,  -1/8, ...
%e A117575    1/1,  0/1,  -1/2,  -1/2,  -1/4,   0/4,   1/8,   1/8, ...
%e A117575   -1/1, -1/2,   0/2,   1/4,   1/4,   1/8,   0/8, -1/16, ...
%e A117575    1/2,  1/2,   1/4,   0/4   -1/8,  -1/8, -1/16,  0/16, ...
%e A117575    0/2, -1/4,  -1/4,  -1/8,   0/8,  1/16,  1/16,  1/32, ...
%e A117575   -1/4,  0/4,   1/8,   1/8,  1/16,  0/16, -1/32, -1/32, ...
%e A117575    1/4,  1/8,   0/8, -1/16, -1/16, -1/32,  0/32,  1/64, ...
%e A117575   -1/8, -1/8, -1/16,  0/16,  1/32,  1/32,  1/64,  0/64. - _Paul Curtz_, Oct 24 2012
%t A117575 CoefficientList[Series[(1-x^3)/((1-x)(1+2x^2)),{x,0,40}],x] (* or *) LinearRecurrence[{0,-2},{1,1,-1},45] (* _Harvey P. Dale_, Apr 09 2018 *)
%o A117575 (PARI) a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ _Charles R Greathouse IV_, Jan 31 2012
%o A117575 (Magma) [1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // _G. C. Greubel_, Apr 19 2023
%o A117575 (SageMath)
%o A117575 def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
%o A117575 [A117575(n) for n in range(51)] # _G. C. Greubel_, Apr 19 2023
%Y A117575 Cf. A016116, A046978, A075553, A116949, A122016, A191754/A192366.
%Y A117575 The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A029744 = {s(n), n>=1}, the numbers 2^k and 3*2^k, as the parent: A029744 (s(n)); A052955 (s(n)-1), A027383 (s(n)-2), A354788 (s(n)-3), A347789 (s(n)-4), A209721 (s(n)+1), A209722 (s(n)+2), A343177 (s(n)+3), A209723 (s(n)+4); A060482, A136252 (minor differences from A354788 at the start); A354785 (3*s(n)), A354789 (3*s(n)-7). The first differences of A029744 are 1,1,1,2,2,4,4,8,8,... which essentially matches eight sequences: A016116, A060546, A117575, A131572, A152166, A158780, A163403, A320770. The bisections of A029744 are A000079 and A007283. - _N. J. A. Sloane_, Jul 14 2022
%K A117575 easy,sign
%O A117575 0,4
%A A117575 _Paul Barry_, Mar 29 2006