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A022410 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=3, a(1)=11.

%I A022410 #30 Sep 08 2022 08:44:46
%S A022410 3,11,15,27,43,71,115,187,303,491,795,1287,2083,3371,5455,8827,14283,
%T A022410 23111,37395,60507,97903,158411,256315,414727,671043,1085771,1756815,
%U A022410 2842587,4599403,7441991,12041395,19483387,31524783,51008171,82532955,133541127
%N A022410 a(n) = a(n-1) + a(n-2) + 1 for n>1, a(0)=3, a(1)=11.
%H A022410 Colin Barker, <a href="/A022410/b022410.txt">Table of n, a(n) for n = 0..1000</a>
%H A022410 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1)
%F A022410 From _R. J. Mathar_, Mar 11 2011: (Start)
%F A022410 a(n+1) - a(n) = A156279(n).
%F A022410 G.f.: (3 + 5*x - 7*x^2) / ((x - 1)*(x^2 + x - 1)).
%F A022410 (End)
%F A022410 a(n) = A156279(n+1) - 1. - _Bruno Berselli_, Jul 27 2017
%F A022410 From _Colin Barker_, Jul 27 2017: (Start)
%F A022410 a(n) = 2^(-n)*(-2^n + 2*(1-sqrt(5))^(1+n) + 2*(1+sqrt(5))^(1+n)).
%F A022410 a(n) = 2*a(n-1) - a(n-3) for n>2.
%F A022410 (End)
%t A022410 LinearRecurrence[{2, 0, -1}, {3, 11, 15}, 40] (* _Bruno Berselli_, Jul 27 2017 *)
%o A022410 (Python)
%o A022410 from sympy import lucas
%o A022410 def a(n): return 4 * lucas(n + 1) - 1
%o A022410 print([a(n) for n in range(51)]) # _Indranil Ghosh_, Jul 27 2017
%o A022410 (PARI) Vec((3+5*x-7*x^2)/((x-1)*(x^2+x-1)) + O(x^50)) \\ _Colin Barker_, Jul 27 2017
%o A022410 (Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!((3+5*x-7*x^2)/((x-1)*(x^2+x-1)))); // _G. C. Greubel_, Feb 28 2018
%Y A022410 Cf. A000032, A156279.
%K A022410 nonn,easy
%O A022410 0,1
%A A022410 _N. J. A. Sloane_