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A045821
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Numerical distance between m-th and (n+m)-th circles in a loxodromic sequence of circles in which each 4 consecutive circles touch.
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1
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-1, 1, 1, 1, 7, 17, 49, 145, 415, 1201, 3473, 10033, 28999, 83809, 242209, 700001, 2023039, 5846689, 16897249, 48833953, 141132743, 407881201, 1178798545, 3406791025, 9845808799, 28454915537, 82236232177, 237667122001
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listen;
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internal format)
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OFFSET
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0,5
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REFERENCES
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Coxeter, H. S. M. "Numerical distances among the circles in a loxodromic sequence." Nieuw Archief voor Wiskunde 16 (1998): 1-10. (Note the word "circles" in the title!)
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LINKS
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FORMULA
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a(n) = 2(a(n-1)+a(n-2)+a(n-3))-a(n-4).
a(n) = Sum{v=0 to [n/2]} binomial(n, 2v)*F(n-v-2) where F(m) is the m-th Fibonacci number.
G.f.: -(x^3-x^2-3*x+1) / (x^4-2*x^3-2*x^2-2*x+1). - Colin Barker, Sep 23 2013
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MAPLE
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with(combinat); F:=fibonacci;
f:=n->add(F(n-i)*binomial(n, 2*(i-2)), i=2..n-1);
[seq(f(n), n=3..32)]; # Produces the sequence from a(3) onwards - N. J. A. Sloane, Sep 03 2018
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MATHEMATICA
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CoefficientList[Series[-(x^3-x^2-3*x+1)/(x^4-2*x^3-2*x^2-2*x+1), {x, 0, 30}], x] (* Stefano Spezia, Sep 12 2018 *)
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PROG
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(PARI) Vec(-(x^3-x^2-3*x+1)/(x^4-2*x^3-2*x^2-2*x+1) + O(x^100)) \\ Colin Barker, Sep 23 2013
(GAP) a:=[-1, 1, 1, 1];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]+2*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Sep 12 2018
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CROSSREFS
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KEYWORD
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sign,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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