A003230 Expansion of 1/((1-x)*(1-2*x)*(1-x-2*x^3)). (Formerly M3417)

1, 4, 11, 28, 67, 152, 335, 724, 1539, 3232, 6727, 13900, 28555, 58392, 118959, 241604, 489459, 989520, 1997015, 4024508, 8100699, 16289032, 32726655, 65705268, 131837763, 264399936, 530028199, 1062139180, 2127809963

Offset

0,2

Comments

The number of simple squares in the (n+4)-th iteration of the Harter-Heighway dragon (see Wikipedia reference below). - Roland Kneer, Jul 01 2013

The number of double points of the (n+4)-th iteration of the Harter-Heighway dragon. - Manfred Lindemann, Nov 11 2015

References

D. E. Daykin and S. J. Tucker, Introduction to Dragon Curves. Unpublished, 1976. See links in A003229 for an earlier version.

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

D. E. Daykin, Letter to N. J. A. Sloane, Dec 1973

D. E. Daykin, Letter to N. J. A. Sloane, Mar 1974

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

Wikipedia, Dragon curve: Harter-Heighway dragon

Index entries for linear recurrences with constant coefficients, signature (4,-5,4,-6,4).

Formula

a(n+3) = a(n+2) + 2*a(n) + 2^(n+4) - 1, with a(-3)=a(-2)=a(-1)=0. - Manfred Lindemann, Nov 11 2015

a(n+2) - a(n+1) = A003477(n+2) + A003477(n). - Manfred Lindemann, Dec 08 2015

a(n) = q(n) + q(n-1) + 2*Sum_{i=0..n-2}(q(i)), where q(i)=A003477 and q(-1)=0. - Manfred Lindemann, Dec 08 2015

From Manfred Lindemann, Nov 11 2015: (Start)

With thrt:=(54+6*sqrt(87))^(1/3), ROR:=(thrt/6-1/thrt) and RORext:=(thrt/6+1/thrt) becomes ROC:=(1/2)*(i*sqrt(3)*RORext-ROR), where i^2=-1.

Now ROR, ROC and conjugate(ROC) are the zeros of 1-x-2*x^3.

With AR:=(2*ROR^2+ROR+2)/(2*ROR-3), AC:=(2*ROC^2+ROC+2)/(2*ROC-3) and the zeros of (1-2*x) and (1-x)

a(n) = (1/2)*(AR*ROR^-(n+4)+AC*ROC^-(n+4)+conjugate(AC*ROC^-(n+4))+1*(1/2)^-(n+4)+1*1^-(n+4)).

Simplified: a(n) = (1/2)*(AR*ROR^-(n+4)+2*Re(AC*ROC^-(n+4))+2^(n+4)+1).

(End)

Maple

A003230:=-1/(z-1)/(2*z-1)/(-1+z+2*z**3); # Simon Plouffe in his 1992 dissertation
S:=series(1/((1-x)*(1-2*x)*(1-x-2*x^3)), x, 101): a:=n->coeff(S, x, n):
seq(a(n), n=0..100); # Manfred Lindemann, Nov 13 2015

Mathematica

CoefficientList[Series[1/((1-x)*(1-2x)*(1-x-2x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2012 *)

Prog

(PARI) Vec(1/((1-x)*(1-2*x)*(1-x-2*x^3))+O(x^66)) \\ Joerg Arndt, Jun 29 2013

Crossrefs

Cf. A003229, A077949, A003477.

Sequence in context: A113478 A056601 A370943 * A099326 A127985 A339252

Adjacent sequences: A003227 A003228 A003229 * A003231 A003232 A003233

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Aug 21 2000

Maple program corrected by Robert Israel, Nov 11 2015

Status

approved