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%I #19 Mar 09 2024 11:02:46
%S 1,4,6,11,18,30,49,80,130,211,342,554,897,1452,2350,3803,6154,9958,
%T 16113,26072,42186,68259,110446,178706,289153,467860,757014,1224875,
%U 1981890,3206766,5188657,8395424
%N a(n) = a(n-1) + a(n-2) + 1, with a(0) = 1 and a(1) = 4.
%H G. C. Greubel, <a href="/A022318/b022318.txt">Table of n, a(n) for n = 0..1000</a>
%H Diana Savin and Elif Tan, <a href="https://arxiv.org/abs/2403.01592">On Companion sequences associated with Leonardo quaternions: Applications over finite fields</a>, arXiv:2403.01592 [math.CO], 2024. See p. 2.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F From _Ron Knott_, Aug 25 2006: (Start)
%F a(n) = 2*A000045(n+2) + A000045(n) - 1.
%F G.f.: (1+2*x-2*x^2)/((1-x)*(1-x-x^2)).
%F a(0)=1, a(1)=4, a(2)=6, a(n) = 2*a(n-1) - a(n-3). (End)
%F a(n) - a(n-1) = A013655(n-1). - _R. J. Mathar_, May 06 2014
%t LinearRecurrence[{2,0,-1}, {1,4,6}, 50] (* _G. C. Greubel_, Aug 25 2017 *)
%o (PARI) x='x+O('x^50); Vec((1+2*x-2*x^2)/((1-x)*(1-x-x^2))) \\ _G. C. Greubel_, Aug 25 2017
%Y Cf. A122195.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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