A006457 Number of elements in Z[ i ] whose 'smallest algorithm' is <= n. (Formerly M3873)

1, 5, 17, 49, 125, 297, 669, 1457, 3093, 6457, 13309, 27201, 55237, 111689, 225101, 452689, 908885, 1822809, 3652701, 7315553, 14645349, 29311081, 58650733, 117342321, 234741877, 469565561, 939245693, 1878655105, 3757539461, 7515406473, 15031271565

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

Cf. A006458, A006459.

Sequence in context: A136303 A268783 A273384 * A115981 A083091 A176953

Adjacent sequences: A006454 A006455 A006456 * A006458 A006459 A006460

Keyword

nonn,easy,nice

Author

H. W. Lenstra, Jr.

Status

approved