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A117575
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Expansion of (1-x^3)/((1-x)*(1+2*x^2)).
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24
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1, 1, -1, -2, 2, 4, -4, -8, 8, 16, -16, -32, 32, 64, -64, -128, 128, 256, -256, -512, 512, 1024, -1024, -2048, 2048, 4096, -4096, -8192, 8192, 16384, -16384, -32768, 32768, 65536, -65536, -131072, 131072, 262144, -262144, -524288, 524288
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OFFSET
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0,4
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COMMENTS
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b(n) = abs(a(n)) = A158780(n+1) = 1,1,1,2,2,4,4,8,8,8,... .
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).
The denominator corresponding to the 0's is a choice.
The classical denominator is 1,1,1,2,1,4,4,8,1,16,16,32,1,... . (End)
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LINKS
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FORMULA
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a(n) = a(n-1) - 2*a(n-2) + 2*a(n-3) for n >= 3.
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).
E.g.f.: (1 + cos(sqrt(2)*x) + sqrt(2)*sin(sqrt(2)*x))/2. - Stefano Spezia, Feb 05 2023
a(n) = (-1)^floor(n/2)*2^floor((n-1)/2), with a(0) = 1. - G. C. Greubel, Apr 19 2023
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EXAMPLE
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0/1, 1/1 1/1, 1/2, 0/2, -1/4, -1/4, -1/8, ...
1/1, 0/1, -1/2, -1/2, -1/4, 0/4, 1/8, 1/8, ...
-1/1, -1/2, 0/2, 1/4, 1/4, 1/8, 0/8, -1/16, ...
1/2, 1/2, 1/4, 0/4 -1/8, -1/8, -1/16, 0/16, ...
0/2, -1/4, -1/4, -1/8, 0/8, 1/16, 1/16, 1/32, ...
-1/4, 0/4, 1/8, 1/8, 1/16, 0/16, -1/32, -1/32, ...
1/4, 1/8, 0/8, -1/16, -1/16, -1/32, 0/32, 1/64, ...
-1/8, -1/8, -1/16, 0/16, 1/32, 1/32, 1/64, 0/64. - Paul Curtz, Oct 24 2012
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MATHEMATICA
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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 *)
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PROG
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(Magma) [1] cat [(-1)^Floor(n/2)*2^Floor((n-1)/2): n in [1..50]]; // G. C. Greubel, Apr 19 2023
(SageMath)
def A117575(n): return 1 if (n==0) else (-1)^(n//2)*2^((n-1)//2)
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CROSSREFS
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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
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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