OFFSET
0,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Robert Israel, Table of n, a(n) for n = 0..3314
Hester Graves, An Elementary Proof of the Minimal Euclidean Function on the Gaussian Integers, arXiv:2205.14043 [math.NT], 2022. See Table 1 p. 25.
H. W. Lenstra, Jr., Letter to N. J. A. Sloane, Nov. 1975
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
P. Samuel, About Euclidean rings, J. Alg., 19 (1971), 282-301.
Index entries for linear recurrences with constant coefficients, signature (4,-3,-6,10,-4).
FORMULA
a(n+5) - 4*a(n+4) + 3*a(n+3) + 6*a(n+2) - 10*a(n+1) + 4*a(n) = 0.
From Robert Israel, Aug 02 2016: (Start)
a(2k) = 14*4^k - 34*2^k + 8*k + 21.
a(2k+1) = 28*4^k - 48*2^k + 8*k + 25.
For n >= 3, a(n) == 5 + 4*n (mod 8). (End)
MAPLE
A006457:=(1+z+2*z^3)/(2*z-1)/(2*z^2-1)/(z-1)^2; # conjectured by Simon Plouffe in his 1992 dissertation
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
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
H. W. Lenstra, Jr.
STATUS
approved