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A003477
Expansion of 1/((1-2x)(1+x^2)(1-x-2x^3)).
(Formerly M2579)
2
1, 3, 6, 14, 33, 71, 150, 318, 665, 1375, 2830, 5798, 11825, 24039, 48742, 98606, 199113, 401455, 808382, 1626038, 3267809, 6562295, 13169814, 26416318, 52962681, 106145855, 212665582, 425965126, 853005201, 1707833095, 3418756806
OFFSET
0,2
COMMENTS
The number of simple squares in the biggest 'cloud' of the Harter-Heighway dragon of degree (n+4). Equals the number of double points in the biggest 'cloud' of the very same. - Manfred Lindemann, Dec 06 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 Unpublished, 1976.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
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
Kevin Ryde, Iterations of the Dragon Curve, see index "BlobA".
FORMULA
a(0) = 1; for n > 0, a(n) = 3*a(n-1) - 3*a(n-2) + 5*a(n-3) - 6*a(n-4) + 2*a(n-5) - 4*a(n-6) (where a(n)=0 for -5 <= n <= -1). - Jon E. Schoenfield, Apr 23 2010
From Manfred Lindemann, Dec 06 2015: (Start)
a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) - 4*a(n-4) + Re(i^(n-4)), a(-5)=a(-4)=a(-3)=a(-2)=0 for all integers n element Z.
a(n+2)+a(n) = A003230(n+2)-A003230(n+1). [Daykin and Tucker equation (5)]
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 BR:=1/(2*ROR-3), BC:=1/(2*ROC-3) and the zeros of (1-2*x) and (1+x^2) becomes
a(n) = (1/2)*(BR*ROR^-(n+4) + BC*ROC^-(n+4) + conjugate(BC*ROC^-(n+4)) + (2/5)*(1/2)^-(n+4) + (3/10 + i*(1/10))*i^-(n+4) + conjugate((3/10 + i*(1/10))*i^-(n+4))).
Simplified: a(n) = (BR/2)*ROR^-(n+4) + Re(BC*ROC^-(n+4)) + (1/5)*(1/2)^-(n+4) + Re((3/10 + i*(1/10))*i^-(n+4)).
(End)
Conjecture: a(n) = A077854(n) + 2*(a(n-3) + a(n-4) + ... + a(1)). - Arie Bos, Nov 29 2019
MAPLE
A003477:=1/(2*z-1)/(-1+z+2*z**3)/(1+z**2); # Simon Plouffe in his 1992 dissertation
S:=series(1/((1-x-2*x^3)*(1-2*x)*(1+x^2)), x, 101): a:=n->coeff(S, x, n):
seq(a(n), n=0..100); # Manfred Lindemann, Dec 06 2015
a:= gfun:-rectoproc({a(n) = 3*a(n-1)-3*a(n-2)+5*a(n-3)-6*a(n-4)+2*a(n-5)-4*a(n-6), seq(a(i)=[1, 3, 6, 14, 33, 71][i+1], i=0..5)}, a(n), remember):
seq(a(n), n=0..100); # Robert Israel, Dec 14 2015
MATHEMATICA
CoefficientList[Series[1/((1-2x)(1+x^2)(1-x-2x^3)), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2012 *)
LinearRecurrence[{3, -3, 5, -6, 2, -4}, {1, 3, 6, 14, 33, 71}, 31] (* Arie Bos, Dec 03 2019 *)
PROG
(PARI) Vec(1/((1-2*x)*(1+x^2)*(1-x-2*x^3))+O(x^99)) \\ Charles R Greathouse IV, Sep 26 2012
CROSSREFS
Cf. A003230, A077949. - Manfred Lindemann, Dec 06 2015
Cf. A077854.
Sequence in context: A330053 A192678 A114945 * A368983 A339600 A078062
KEYWORD
nonn,easy
EXTENSIONS
More terms from Jon E. Schoenfield, Apr 23 2010
STATUS
approved