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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 (list; graph; refs; listen; history; text; internal format)
 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 Unpublished, 1976.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 Vincenzo Librandi, Table of n, a(n) for n = 0..1000 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. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Index entries for linear recurrences with constant coefficients, signature (3,-3,5,-6,2,-4). 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 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. - Manfred Lindemann, Dec 06 2015 a(n+2)+a(n) = A003230(n+2)-A003230(n+1). - Manfred Lindemann, Dec 06 2015 From Manfred Lindemann, Dec 06 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 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 3, 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 * A078062 A275873 A018017 Adjacent sequences:  A003474 A003475 A003476 * A003478 A003479 A003480 KEYWORD nonn,easy AUTHOR EXTENSIONS More terms from Jon E. Schoenfield, Apr 23 2010 STATUS approved

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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)