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%I #20 Sep 08 2022 08:44:51
%S 3,5,9,14,22,34,53,83,131,208,332,532,855,1377,2221,3586,5794,9366,
%T 15145,24495,39623,64100,103704,167784,271467,439229,710673,1149878,
%U 1860526,3010378,4870877,7881227,12752075,20633272,33385316,54018556,87403839,141422361,228826165,370248490
%N Convolution of natural numbers n >= 1 with Lucas numbers L(k) for k >= -2.
%H G. C. Greubel, <a href="/A033818/b033818.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F a(n) = L(1)*(F(n+1) - 1) + L(0)*F(n) - L(-1)*n, F(n): Fibonacci (A000045), L(n): Lucas (A000032) with L(-n) = (-1)^n*L(n).
%F G.f.: x*(3-4*x)/((1-x-x^2)*(1-x)^2).
%F a(n) = Lucas(n+1) + n - 1. - _G. C. Greubel_, Jun 01 2019
%t LinearRecurrence[{3,-2,-1,1}, {3,5,9,14}, 50] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2011, modified by _G. C. Greubel_, Jun 01 2019 *)
%t Table[LucasL[n+1] +n-1, {n,1,50}] (* _G. C. Greubel_, Jun 01 2019 *)
%o (PARI) {a(n) = fibonacci(n+2) + fibonacci(n) + n-1}; \\ _G. C. Greubel_, Jun 01 2019
%o (Magma) [Lucas(n+1) +n-1: n in [1..50]]; // _G. C. Greubel_, Jun 01 2019
%o (Sage) [lucas_number2(n+1,1,-1) +n-1 for n in (1..50)] # _G. C. Greubel_, Jun 01 2019
%o (GAP) List([1..50], n-> Lucas(1,-1,n+1)[2] +n-1) # _G. C. Greubel_, Jun 01 2019
%Y Cf. A000032, A000045, A023537, A023548, A033811.
%K easy,nonn
%O 1,1
%A _Wolfdieter Lang_
%E Terms a(31) onward added by _G. C. Greubel_, Jun 01 2019
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