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%I #29 Sep 13 2018 09:51:02
%S -1,1,1,1,7,17,49,145,415,1201,3473,10033,28999,83809,242209,700001,
%T 2023039,5846689,16897249,48833953,141132743,407881201,1178798545,
%U 3406791025,9845808799,28454915537,82236232177,237667122001
%N Numerical distance between m-th and (n+m)-th circles in a loxodromic sequence of circles in which each 4 consecutive circles touch.
%D 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!)
%H Muniru A Asiru, <a href="/A045821/b045821.txt">Table of n, a(n) for n = 0..500</a>
%H H. S. M. Coxeter, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002022109">Loxodromic sequences of tangent spheres</a>, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.
%H H. S. M. Coxeter, <a href="https://doi.org/10.1007/BF03024413">Numerical distances among the spheres in a loxodromic sequence</a>, Math. Intell. 19(4) 1997 pp. 41-47. (Note the word "spheres" in the title!) See page 45.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F a(n) = 2(a(n-1)+a(n-2)+a(n-3))-a(n-4).
%F a(n) = Sum{v=0 to [n/2]} binomial(n, 2v)*F(n-v-2) where F(m) is the m-th Fibonacci number.
%F 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
%F Lim_{n -> inf} a(n)/a(n-1) = A318605. - _A.H.M. Smeets_, Sep 12 2018
%p with(combinat); F:=fibonacci;
%p f:=n->add(F(n-i)*binomial(n,2*(i-2)), i=2..n-1);
%p [seq(f(n),n=3..32)]; # Produces the sequence from a(3) onwards - _N. J. A. Sloane_, Sep 03 2018
%t 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 *)
%o (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
%o (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
%Y Cf. A027674.
%K sign,easy
%O 0,5
%A _Colin Mallows_
%E Reference and formulas from _Floor van Lamoen_