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, 7, 17, 49, 145, 415, 1201, 3473, 10033, 28999, 83809, 242209, 700001, 2023039, 5846689, 16897249, 48833953, 141132743, 407881201, 1178798545, 3406791025, 9845808799, 28454915537, 82236232177, 237667122001

Offset: 0

Author

Colin Mallows

Keywords

• sign
• easy

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).

Crossrefs

Cf. A027674.

Programs

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

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 *)

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

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

Extensions

Reference and formulas from Floor van Lamoen