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

Author

N. J. A. Sloane, Sep 03 2018

Keywords

• nonn
• easy

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).

Programs

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 *)

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

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

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