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A003477
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Expansion of 1/((1-2x)(1+x^2)(1-x-2x^3)).
(Formerly M2579)
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2
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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 Unpublished, 1976.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).
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LINKS
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FORMULA
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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.
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
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MAPLE
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S:=series(1/((1-x-2*x^3)*(1-2*x)*(1+x^2)), x, 101): a:=n->coeff(S, x, n):
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):
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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