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%I #57 Apr 20 2023 02:10:59
%S 1,1,-1,-2,2,4,-4,-8,8,16,-16,-32,32,64,-64,-128,128,256,-256,-512,
%T 512,1024,-1024,-2048,2048,4096,-4096,-8192,8192,16384,-16384,-32768,
%U 32768,65536,-65536,-131072,131072,262144,-262144,-524288,524288
%N Expansion of (1-x^3)/((1-x)*(1+2*x^2)).
%C Row sums of A116949.
%C From _Paul Curtz_, Oct 24 2012: (Start)
%C b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
%C 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 The denominator corresponding to the 0's is a choice.
%C The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)
%H G. C. Greubel, <a href="/A117575/b117575.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,-2).
%F a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
%F 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 a(n+1) = Sum_{k=0..n} A122016(n,k)*(-1)^k. - _Philippe Deléham_, Jan 31 2012
%F E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - _Stefano Spezia_, Feb 05 2023
%F a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - _G. C. Greubel_, Apr 19 2023
%e 0/1, 1/1 1/1, 1/2, 0/2, -1/4, -1/4, -1/8, ...
%e 1/1, 0/1, -1/2, -1/2, -1/4, 0/4, 1/8, 1/8, ...
%e -1/1, -1/2, 0/2, 1/4, 1/4, 1/8, 0/8, -1/16, ...
%e 1/2, 1/2, 1/4, 0/4 -1/8, -1/8, -1/16, 0/16, ...
%e 0/2, -1/4, -1/4, -1/8, 0/8, 1/16, 1/16, 1/32, ...
%e -1/4, 0/4, 1/8, 1/8, 1/16, 0/16, -1/32, -1/32, ...
%e 1/4, 1/8, 0/8, -1/16, -1/16, -1/32, 0/32, 1/64, ...
%e -1/8, -1/8, -1/16, 0/16, 1/32, 1/32, 1/64, 0/64. - _Paul Curtz_, Oct 24 2012
%t 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 (PARI) a(n)=if(n,(-1)^(n\2)<<((n-1)\2),1) \\ _Charles R Greathouse IV_, Jan 31 2012
%o (Magma) [1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // _G. C. Greubel_, Apr 19 2023
%o (SageMath)
%o def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
%o [A117575(n) for n in range(51)] # _G. C. Greubel_, Apr 19 2023
%Y Cf. A016116, A046978, A075553, A116949, A122016, A191754/A192366.
%Y 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 easy,sign
%O 0,4
%A _Paul Barry_, Mar 29 2006
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