This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A081067 #38 Jul 02 2025 16:02:01
%S A081067 5,20,125,845,5780,39605,271445,1860500,12752045,87403805,599074580,
%T A081067 4106118245,28143753125,192900153620,1322157322205,9062201101805,
%U A081067 62113250390420,425730551631125,2918000611027445,20000273725560980
%N A081067 a(n) = Lucas(4n+2)+2, or 5*Fibonacci(2n+1)^2.
%C A081067 a(n) is the square of limit of (G(j+2n-1) + G(j-2n+1))/G(j) as j -> infinity, where G(n) is any sequence of the form G(n+1) = G(n) + G(n-1), with any starting values, including non-integer values. G(n) includes Lucas and Fibonacci. Compare with A005248 for even number offsets from j in any such G(n). - _Richard R. Forberg_, Nov 16 2014
%C A081067 a(n) = (t(i+6n+3) + t(i))/(t(i+4n+2) + t(i+2n+1)) + 3, where (t) is any sequence of the form t(n+2) = 2t(n+1) + 2t(n) - t(n-1) or of the form t(n+1) = 3t(n) - t(n-1) without regard to initial values as long as t(i+4n+2) + t(i+2n+1) != 0. - _Klaus Purath_, Jun 23 2024
%D A081067 Hugh C. Williams, Edouard Lucas and Primality Testing, John Wiley and Sons, 1998, p. 75.
%H A081067 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (8,-8,1).
%F A081067 a(n) = 8a(n-1) - 8a(n-2) + a(n-3).
%F A081067 G.f.: -5*(x^2-4*x+1)/((x-1)*(x^2-7*x+1)). - _Colin Barker_, Jun 25 2012
%F A081067 a(n) ~ phi^(4n+2). - _Charles R Greathouse IV_, Nov 17 2014
%F A081067 a(n) = 5*A081068(n). - _R. J. Mathar_, Feb 13 2016
%p A081067 luc := proc(n) option remember: if n=0 then RETURN(2) fi: if n=1 then RETURN(1) fi: luc(n-1)+luc(n-2): end: for n from 0 to 40 do printf(`%d,`,luc(4*n+2)+2) od: # _James Sellers_, Mar 05 2003
%t A081067 Table[LucasL[4n+2]+2,{n,0,20}] (* or *)
%t A081067 Table[5Fibonacci[2n+1]^2,{n,0,30}] (* _Harvey P. Dale_, Apr 18 2011 *)
%o A081067 (PARI) a(n)=5*fibonacci(2*n+1)^2 \\ _Charles R Greathouse IV_, Nov 17 2014
%Y A081067 Cf. A000045 (Fibonacci numbers), A000032 (Lucas numbers).
%K A081067 nonn,easy
%O A081067 0,1
%A A081067 _R. K. Guy_, Mar 04 2003