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A065034 a(n) = Lucas(2*n) + 1.

%I A065034 #60 May 29 2025 16:34:17
%S A065034 3,4,8,19,48,124,323,844,2208,5779,15128,39604,103683,271444,710648,
%T A065034 1860499,4870848,12752044,33385283,87403804,228826128,599074579,
%U A065034 1568397608,4106118244,10749957123,28143753124,73681302248,192900153619,505019158608,1322157322204
%N A065034 a(n) = Lucas(2*n) + 1.
%H A065034 Harry J. Smith, <a href="/A065034/b065034.txt">Table of n, a(n) for n = 0..200</a>
%H A065034 Sela Fried, <a href="https://arxiv.org/abs/2505.14196">Even-up words and their variants</a>, arXiv:2505.14196 [math.CO], 2025. See p. 8.
%H A065034 Markus Grassl and Andrew J. Scott, <a href="https://arxiv.org/abs/1707.02944">Fibonacci-Lucas SIC-POVMs</a>, arXiv:1707.02944 [quant-ph], 2017.
%H A065034 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F A065034 a(n) = F(2*n+1) + F(2*n-1) + 1 = A005248(n) + 1.
%F A065034 From _R. J. Mathar_, Jul 18 2009: (Start)
%F A065034 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3).
%F A065034 G.f.: 1/(1-x) + (2-3*x)/(1-3*x+x^2). (End)
%F A065034 a(n) = 1 + ((3-sqrt(5))/2)^n + ((3+sqrt(5))/2)^n. - _Colin Barker_, Nov 01 2016
%p A065034 a:= n-> (<<0|1>, <1|1>>^(2*n). <<2,1>>)[1, 1]+1:
%p A065034 seq(a(n), n=0..30);  # _Alois P. Heinz_, Nov 01 2016
%t A065034 LucasL[2 Range[30]]+1 (* _Harvey P. Dale_, Oct 21 2011 *)
%t A065034 LinearRecurrence[{4, -4, 1}, {3, 4, 8}, 30] (* _Jean-François Alcover_, Jan 08 2019 *)
%o A065034 (PARI) a(n) = { fibonacci(2*n + 1) + fibonacci(2*n - 1) + 1 } \\ _Harry J. Smith_, Oct 03 2009
%o A065034 (PARI) Vec((3-2*x)*(1-2*x)/((1-x)*(1-3*x+x^2)) + O(x^40)) \\ _Colin Barker_, Nov 01 2016
%o A065034 (Magma) [ Lucas(2*n) + 1: n in [0..210]]; // _Vincenzo Librandi_, Apr 15 2011
%Y A065034 Cf. A000032, A000045, A005248, A005592.
%Y A065034 Cf. A002878 (first differences). - _R. J. Mathar_, Jul 18 2009
%K A065034 nonn,easy
%O A065034 0,1
%A A065034 _N. J. A. Sloane_, Nov 04 2001
%E A065034 a(0)=3 prepended by _Joerg Arndt_, Nov 01 2016