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%I M3873 #49 May 30 2022 02:26:49
%S 1,5,17,49,125,297,669,1457,3093,6457,13309,27201,55237,111689,225101,
%T 452689,908885,1822809,3652701,7315553,14645349,29311081,58650733,
%U 117342321,234741877,469565561,939245693,1878655105,3757539461,7515406473,15031271565
%N Number of elements in Z[ i ] whose 'smallest algorithm' is <= n.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Robert Israel, <a href="/A006457/b006457.txt">Table of n, a(n) for n = 0..3314</a>
%H Hester Graves, <a href="https://arxiv.org/abs/2205.14043">An Elementary Proof of the Minimal Euclidean Function on the Gaussian Integers</a>, arXiv:2205.14043 [math.NT], 2022. See Table 1 p. 25.
%H H. W. Lenstra, Jr., <a href="/A006457/a006457.pdf">Letter to N. J. A. Sloane, Nov. 1975</a>
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H P. Samuel, <a href="http://dx.doi.org/10.1016/0021-8693(71)90110-4">About Euclidean rings</a>, J. Alg., 19 (1971), 282-301.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-6,10,-4).
%F a(n+5) - 4*a(n+4) + 3*a(n+3) + 6*a(n+2) - 10*a(n+1) + 4*a(n) = 0.
%F From _Robert Israel_, Aug 02 2016: (Start)
%F a(2k) = 14*4^k - 34*2^k + 8*k + 21.
%F a(2k+1) = 28*4^k - 48*2^k + 8*k + 25.
%F For n >= 3, a(n) == 5 + 4*n (mod 8). (End)
%p A006457:=(1+z+2*z^3)/(2*z-1)/(2*z^2-1)/(z-1)^2; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%p seq(op([14*4^k-34*2^k+8*k+21, 28*4^k-48*2^k+8*k+25]), k=0..50); # _Robert Israel_, Aug 02 2016
%t CoefficientList[Series[(1+x+2x^3)/(2x-1)/(2x^2-1)/(x-1)^2,{x,0,30}], x] (* or *) LinearRecurrence[{4,-3,-6,10,-4},{1,5,17,49,125},30] (* _Harvey P. Dale_, Jun 22 2011 *)
%Y Cf. A006458, A006459.
%K nonn,easy,nice
%O 0,2
%A H. W. Lenstra, Jr.
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