cp's OEIS Frontend

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.

%I A317974 #33 Feb 16 2025 08:33:56
%S A317974 0,0,1,1,4,12,33,97,280,808,2337,6753,19516,56404,163009,471105,
%T A317974 1361520,3934864,11371969,32865601,94983348,274506972,793339873,
%U A317974 2292794785,6626299912,19150362168,55345573857,159951677089,462268926316,1335981992356,3861059617665
%N 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.
%H A317974 A.H.M. Smeets, <a href="/A317974/b317974.txt">Table of n, a(n) for n = 0..2172</a>
%H A317974 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 A317974 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CoxetersLoxodromicSequenceofTangentCircles.html">Coxeter's Loxodromic Sequence of Tangent Circles</a>
%H A317974 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,2,-1).
%F A317974 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
%F A317974 G.f.: x^2*(1 - x) / (1 - 2*x - 2*x^2 - 2*x^3 + x^4). - _Colin Barker_, Sep 04 2018
%t A317974 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 *)
%o A317974 (Python)
%o A317974 a1,a2,a3,a4,n = 1,1,0,0,3
%o A317974 print(0,0)
%o A317974 print(1,0)
%o A317974 print(2,1)
%o A317974 print(3,1)
%o A317974 while n < 2172:
%o A317974     a1,a2,a3,a4,n = 2*(a1+a2+a3)-a4,a1,a2,a3,n+1
%o A317974     print(n,a1) # _A.H.M. Smeets_, Sep 04 2018
%o A317974 (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
%K A317974 nonn,easy
%O A317974 0,5
%A A317974 _N. J. A. Sloane_, Sep 03 2018