A006458 Number of elements in Z[ omega ] whose 'smallest algorithm' is <= n, where omega^2 = -omega - 1. (Formerly M4399)

1, 7, 31, 115, 391, 1267, 3979, 12271, 37423, 113371, 342091, 1029799, 3095671, 9298147, 27914179, 83777503, 251394415, 754292827, 2263072411, 6789560407, 20369288455, 61108939795, 183328720435, 549989524879, 1649974525855, 4949934107083, 14849820951115

Offset

0,2

Comments

omega is a primitive third root of unity. - Joerg Arndt, Apr 29 2021

References

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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 (5,-5,-5,4,8,-6).

Formula

a(n+6) - 5*a(n+5) + 5*a(n+4) + 5*a(n+3) - 4*a(n+2) - 8*a(n+1) + 6*a(n) = 0.

G.f.: (x*(6*x^4+2*x^3+x+2)+1)/((x-1)^2*(3*x-1)*(2*x^2*(x+1)-1)). - Harvey P. Dale, Mar 03 2012

Maple

A006458:=(1+2*z+z**2+2*z**4+6*z**5)/(3*z-1)/(2*z**3+2*z**2-1)/(z-1)**2; # Conjectured by Simon Plouffe in his 1992 dissertation

Mathematica

LinearRecurrence[{5, -5, -5, 4, 8, -6}, {1, 7, 31, 115, 391, 1267}, 40] (* Harvey P. Dale, Mar 03 2012 *)

Crossrefs

Cf. A006457, A006459.

Sequence in context: A097786 A350498 A197649 * A091344 A032197 A114289

Adjacent sequences: A006455 A006456 A006457 * A006459 A006460 A006461

Keyword

nonn,easy,nice

Author

H. W. Lenstra, Jr.

Extensions

Corrected by T. D. Noe, Nov 08 2006

More terms from Harvey P. Dale, Mar 03 2012

Name corrected by Joerg Arndt, Apr 29 2021

Status

approved