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%I #63 Mar 08 2024 12:00:12
%S 4,29,199,1364,9349,64079,439204,3010349,20633239,141422324,969323029,
%T 6643838879,45537549124,312119004989,2139295485799,14662949395604,
%U 100501350283429,688846502588399,4721424167835364,32361122672259149,221806434537978679,1520283919093591604
%N a(n) = Lucas(4*n + 3).
%C a(n) mod 4 gives A101000.
%H Colin Barker, <a href="/A288913/b288913.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 G.f.: (4 + x)/(1 - 7*x + x^2).
%F a(n) = 7*a(n-1) - a(n-2) for n>1, with a(0)=4, a(1)=29.
%F a(n) = ((sqrt(5) + 1)^(4*n + 3) - (sqrt(5) - 1)^(4*n + 3))/(8*16^n).
%F a(n) = Fibonacci(4*n+4) + Fibonacci(4*n+2).
%F a(n) = 4*A004187(n+1) + A004187(n).
%F a(n) = 5*A003482(n) + 4 = 5*A081016(n) - 1.
%F a(n) = A002878(2*n+1) = A093960(2*n+3) = A001350(4*n+3) = A068397(4*n+3).
%F a(n+1)*a(n+k) - a(n)*a(n+k+1) = 15*Fibonacci(4*k). Example: for k=6, a(n+1)*a(n+6) - a(n)*a(n+7) = 15*Fibonacci(24) = 695520.
%t LucasL[4 Range[0, 21] + 3]
%t LinearRecurrence[{7,-1}, {4,29}, 30] (* _G. C. Greubel_, Dec 22 2017 *)
%o (PARI) Vec((4 + x)/(1 - 7*x + x^2) + O(x^30)) \\ _Colin Barker_, Jun 20 2017
%o (Sage)
%o def L():
%o x, y = -1, 4
%o while True:
%o yield y
%o x, y = y, 7*y - x
%o r = L(); [next(r) for _ in (0..21)] # _Peter Luschny_, Jun 20 2017
%o (Magma) [Lucas(4*n + 3): n in [0..30]]; // _G. C. Greubel_, Dec 22 2017
%o (Python)
%o from sympy import lucas
%o def a(n): return lucas(4*n + 3)
%o print([a(n) for n in range(22)]) # _Michael S. Branicky_, Apr 29 2021
%Y Cf. A000032, A000045, A004187, A101000.
%Y Cf. A033891: fourth quadrisection of A000045.
%Y Partial sums are in A081007 (after 0).
%Y Positive terms of A098149, and subsequence of A001350, A002878, A016897, A093960, A068397.
%Y Quadrisection of A000032: A056854 (first), A056914 (second), A246453 (third, without 11), this sequence (fourth).
%K nonn,easy
%O 0,1
%A _Bruno Berselli_, Jun 19 2017
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