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Revision History for A171663

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A171663

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)).
(history; published version)

#24 by Charles R Greathouse IV at Thu Sep 08 08:45:50 EDT 2022

PROG

(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

Discussion

Thu Sep 08

08:45

OEIS Server: https://oeis.org/edit/global/2944

#23 by Michel Marcus at Mon Jun 03 00:58:15 EDT 2019

STATUS

reviewed

approved

#22 by Joerg Arndt at Mon Jun 03 00:54:00 EDT 2019

STATUS

proposed

reviewed

#21 by Joerg Arndt at Mon Jun 03 00:53:50 EDT 2019

STATUS

editing

proposed

#20 by Joerg Arndt at Mon Jun 03 00:53:31 EDT 2019

NAME

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)).

COMMENTS

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, ...

EXTENSIONS

New name from Joerg Arndt, Jun 03 2019

STATUS

proposed

editing

Discussion

Mon Jun 03

00:53

Joerg Arndt: Plagiarism deleted.

#19 by G. C. Greubel at Sat Jun 01 03:07:51 EDT 2019

STATUS

editing

proposed

Discussion

Sat Jun 01

09:43

Joerg Arndt: ..and the comments.

13:29

Jon E. Schoenfield: I think the existing Name is misleading. At first, I took the words "plus or minus" to apply to the remainder of the Name, which would make the Name mean the same thing as

   2^(2*n+1) +- (2^(n+1)+1)

but that would give terms including 3, 23, 111, ..., which are not in the Data section.  Then I realized that the "plus or minus" was intended to apply only to the "2^(n+1)" part, not to the "2(n+1)+1".

Since 5 is listed twice, the Name can't be changed to something beginning with "Numbers of the form" ....

Could we change the Name to either

   a(n) = 2^(2*k+1) - 2^(k+1) + 1 for even n, 2^(2*k+1) + 2^(k+1) + 1 for odd n, where k = floor(n/2).
or
   a(n) = 2^(2*k+1) -(-1)^n*2^(k+1) + 1, where k = floor(n/2).
?

13:42

Jon E. Schoenfield: @Joerg -- yes, I have real problems with the Comments section!  Taken at face value, that section seems to indicate that the "we" includes the author of this sequence.

Apparently, however, the author of this sequence merely copied (without giving any indication that the words were not his own) the abstract from the paper by Tsumura (probably taking the form of it that's given at the page at the link, i.e., with "^" characters rather than superscripts, and replacing "\pm" there with "plus or minus") ... and then -- without so much as a paragraph break after the "We propose ..." sentence -- appended his own sentence ("I list '5' twice ...").

#18 by G. C. Greubel at Sat Jun 01 03:07:32 EDT 2019

LINKS

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.

FORMULA

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

MATHEMATICA

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 *)

PROG

(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

STATUS

proposed

editing

#17 by Michel Marcus at Sat Jun 01 01:47:30 EDT 2019

STATUS

editing

proposed

#16 by Michel Marcus at Sat Jun 01 01:47:25 EDT 2019

LINKS

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).

STATUS

proposed

editing

#15 by Jon E. Schoenfield at Fri May 31 23:26:43 EDT 2019

STATUS

editing

proposed