(MAGMAMagma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2)) )); // G. C. Greubel, Jun 01 2019
(MAGMAMagma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2)) )); // G. C. Greubel, Jun 01 2019
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2^(2*n+1) plus or minus 2^(n+1)+1.
Expansion of (1 + 4*x - 6*x^2 - 16*x^3 + 20*x^4)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)).
Robert Denomme and Gordan Savin made a primality test for Fermat numbers 2^(2^k)+1 using elliptic curves. We propose another primality test using elliptic curves for Fermat numbers and also give primality tests for integers of the form 2^(2k+1) plus or minus 2^(k+1)+1. I list "5" twice, as it, uniquely, occurs twice. The primes in this sequence begin: 5, 5, 13, 41, 113, 2113, ...
New name from Joerg Arndt, Jun 03 2019
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G. C. Greubel, <a href="/A171663/b171663.txt">Table of n, a(n) for n = 0..1000</a>
Yu Tsumura, <a href="http://arxiv.org/abs/0912.2116">Primality tests for Fermat numbers and 2^(2k+1)\pm2 +/- 2^(k+1)+1</a>, arXiv:0912.2116 [math.NT], Dec 10 2009.
G.f.: -(201 + 4*x - 6*x^4 2 - 16*x^3 - 6+ 20*x^2 + 4*x + )/((1) / ((-x-)*(1)*(-2*x-)*(1)*(+2*x+)*(1)*(-2*x^2-1)). - Colin Barker, Apr 27 2013
Flatten[Table[2^(2*n + 1) + 1 + 2^(n + 1) {-1, 1}, {n, 0, 10040}]] (* J. Mulder (jasper.mulder(AT)planet.nl), Jan 28 2010 *)
(PARI) my(x='x+O('x^40)); Vec((1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2))) \\ G. C. Greubel, Jun 01 2019
(MAGMA) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2)) )); // G. C. Greubel, Jun 01 2019
(Sage) ((1+4*x-6*x^2-16*x^3+20*x^4)/((1-x)*(1- 2*x^2)*(1-4*x^2))).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jun 01 2019
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Yu Tsumura, <a href="http://arxiv.org/abs/0912.2116">Primality tests for Fermat numbers and 2^(2k+1)\pm2^(k+1)+1</a>, arXiv:0912.2116 [math.NT], Dec 10, 2009.
<a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,6,-6,-8,8).
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