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%I #31 Sep 08 2022 08:45:01
%S 1,11,76,521,3571,24476,167761,1149851,7881196,54018521,370248451,
%T 2537720636,17393796001,119218851371,817138163596,5600748293801,
%U 38388099893011,263115950957276,1803423556807921,12360848946698171
%N a(n) = Lucas(4*n+1).
%D V. E. Hoggatt, Jr., Fibonacci and Lucas Numbers, A Publication of the Fibonacci Association, Houghton Mifflin Co., 1969, pp. 27-29.
%H G. C. Greubel, <a href="/A056914/b056914.txt">Table of n, a(n) for n = 0..1000</a>
%H Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (7,-1).
%F a(n) = 7*a(n-1) - a(n-2), with a(0)=1, a(1)=11.
%F 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 G.f.: (1+4*x)/(1-7*x+x^2). - _Philippe Deléham_, Nov 02 2008
%p with(combinat); seq(fibonacci(4*n+2)+fibonacci(4*n), n = 0..30); # _G. C. Greubel_, Jan 16 2020
%t LucasL[4*Range[0,30]+1] (* or *) LinearRecurrence[{7,-1}, {1,11}, 30] (* _G. C. Greubel_, Dec 24 2017 *)
%o (PARI) my(x='x+O('x^30)); Vec((1+4*x)/(1-7*x+x^2)) \\ _G. C. Greubel_, Dec 24 2017
%o (Magma) [Lucas(4*n+1): n in [0..30]]; // _G. C. Greubel_, Dec 24 2017
%o (Sage) [lucas_number2(4*n+1,1,-1) for n in (0..30)] # _G. C. Greubel_, Jan 16 2020
%o (GAP) List([0..30], n-> Lucas(1,-1,4*n+1)[2] ); # _G. C. Greubel_, Jan 16 2020
%Y Cf. (A056914) = sqrt{5*(A033889)^2 - 4}.
%Y Cf. quadrisection of A000032: A056854 (first), this sequence (second), A246453 (third, without 11), A288913 (fourth).
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jul 11 2000
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