A317974 - OEIS

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

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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