A317974 a(n) = 2*(a(n-1)+a(n-2)+a(n-3))-a(n-4) for n >= 4, with initial terms 0,0,1,1.

0, 0, 1, 1, 4, 12, 33, 97, 280, 808, 2337, 6753, 19516, 56404, 163009, 471105, 1361520, 3934864, 11371969, 32865601, 94983348, 274506972, 793339873, 2292794785, 6626299912, 19150362168, 55345573857, 159951677089, 462268926316, 1335981992356, 3861059617665

Offset

0,5

Links

A.H.M. Smeets, Table of n, a(n) for n = 0..2172

H. S. M. Coxeter, Loxodromic sequences of tangent spheres, Aequationes Mathematicae, 1.1-2 (1968): 104-121. See p. 112.

Eric Weisstein's World of Mathematics, Coxeter's Loxodromic Sequence of Tangent Circles

Index entries for linear recurrences with constant coefficients, signature (2,2,2,-1).

Formula

Lim {n -> infinity} log(a(n))/n = 1.0612750619050... = log(phi+sqrt(phi)) = log(A001622+A139339), where phi is the golden ratio. - A.H.M. Smeets, Sep 04 2018

G.f.: x^2*(1 - x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - Colin Barker, Sep 04 2018

Mathematica

nxt[{a_, b_, c_, d_}]:={b, c, d, 2(b+c+d)-a}; NestList[nxt, {0, 0, 1, 1}, 30][[;; , 1]] (* or *) LinearRecurrence[{2, 2, 2, -1}, {0, 0, 1, 1}, 40] (* Harvey P. Dale, Dec 10 2024 *)

Prog

(Python)
a1, a2, a3, a4, n = 1, 1, 0, 0, 3
print(0, 0)
print(1, 0)
print(2, 1)
print(3, 1)
while n < 2172:
    a1, a2, a3, a4, n = 2*(a1+a2+a3)-a4, a1, a2, a3, n+1
    print(n, a1) # A.H.M. Smeets, Sep 04 2018
(PARI) concat(vector(2), Vec(x^2*(1 - x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4) + O(x^40))) \\ Colin Barker, Sep 04 2018

Crossrefs

Sequence in context: A135254 A326804 A000754 * A119683 A331834 A135373

Adjacent sequences: A317971 A317972 A317973 * A317975 A317976 A317977

Keyword

nonn,easy,changed

Author

N. J. A. Sloane, Sep 03 2018

Status

approved