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 A056914 #36 Jul 19 2024 11:31:13
%S A056914 1,11,76,521,3571,24476,167761,1149851,7881196,54018521,370248451,
%T A056914 2537720636,17393796001,119218851371,817138163596,5600748293801,
%U A056914 38388099893011,263115950957276,1803423556807921,12360848946698171
%N A056914 a(n) = Lucas(4*n+1).
%C A056914 a(n) = (t(i+4n+1) - t(i))/(t(i+2n+1) - t(i+2n)), where (t) is any sequence of the form t(n+2) = 4t(n+1) - 4t(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+2n+1) - t(i+2n) != 0. - _Klaus Purath_, Jun 24 2024
%D A056914 V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers, A Publication of the Fibonacci Association, Houghton Mifflin Co., 1969, pp. 27-29.
%H A056914 G. C. Greubel, <a href="/A056914/b056914.txt">Table of n, a(n) for n = 0..1000</a>
%H A056914 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A056914 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%F A056914 a(n) = 7*a(n-1) - a(n-2), with a(0)=1, a(1)=11.
%F A056914 a(n) = (11*(((7+3*sqrt(5))/2)^n - ((7-3*sqrt(5))/2)^n) - (((7+3*sqrt(5))/2)^(n-1) - ((7-3*sqrt(5))/2)^(n-1)))/3*sqrt(5).
%F A056914 G.f.: (1+4*x)/(1-7*x+x^2). - _Philippe Deléham_, Nov 02 2008
%p A056914 with(combinat); seq(fibonacci(4*n+2)+fibonacci(4*n), n = 0..30); # _G. C. Greubel_, Jan 16 2020
%t A056914 LucasL[4*Range[0,30]+1] (* or *) LinearRecurrence[{7,-1}, {1,11}, 30] (* _G. C. Greubel_, Dec 24 2017 *)
%o A056914 (PARI) my(x='x+O('x^30)); Vec((1+4*x)/(1-7*x+x^2)) \\ _G. C. Greubel_, Dec 24 2017
%o A056914 (Magma) [Lucas(4*n+1): n in [0..30]]; // _G. C. Greubel_, Dec 24 2017
%o A056914 (Sage) [lucas_number2(4*n+1,1,-1) for n in (0..30)] # _G. C. Greubel_, Jan 16 2020
%o A056914 (GAP) List([0..30], n-> Lucas(1,-1,4*n+1)[2] ); # _G. C. Greubel_, Jan 16 2020
%Y A056914 Cf. (A056914) = sqrt{5*(A033889)^2 - 4}.
%Y A056914 Cf. quadrisection of A000032: A056854 (first), this sequence (second), A246453 (third, without 11), A288913 (fourth).
%K A056914 easy,nonn
%O A056914 0,2
%A A056914 _Barry E. Williams_, Jul 11 2000