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 A045821 Numerical distance between m-th and (n+m)-th circles in a loxodromic sequence of circles in which each 4 consecutive circles touch. 1
 -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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 REFERENCES 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!) LINKS Muniru A Asiru, Table of n, a(n) for n = 0..500 H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112. H. S. M. Coxeter, Numerical distances among the spheres in a loxodromic sequence, Math. Intell. 19(4) 1997 pp. 41-47. (Note the word "spheres" in the title!) See page 45. Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1). FORMULA 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 Lim_{n -> inf} a(n)/a(n-1) = A318605. - A.H.M. Smeets, Sep 12 2018 MAPLE 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 MATHEMATICA 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 *) PROG (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 CROSSREFS Cf. A027674. Sequence in context: A179262 A018672 A221591 * A262754 A115914 A352616 Adjacent sequences: A045818 A045819 A045820 * A045822 A045823 A045824 KEYWORD sign,easy AUTHOR Colin Mallows EXTENSIONS Reference and formulas from Floor van Lamoen STATUS approved

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