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A306672 Partial sums of the even Lucas numbers (A014448).

%I A306672 #36 Jan 18 2021 15:16:09
%S A306672 2,6,24,100,422,1786,7564,32040,135722,574926,2435424,10316620,
%T A306672 43701902,185124226,784198804,3321919440,14071876562,59609425686,
%U A306672 252509579304,1069647742900,4531100550902,19194049946506,81307300336924,344423251294200,1459000305513722,6180424473349086
%N A306672 Partial sums of the even Lucas numbers (A014448).
%H A306672 Robert Israel, <a href="/A306672/b306672.txt">Table of n, a(n) for n = 0..1593</a>
%H A306672 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-3,-1).
%F A306672 a(n) = L(0) + L(3) + L(6) + L(9) +  ... + L(3n), L(n) = Lucas numbers A000032.
%F A306672 a(n) = Sum_{i=0..n} L(3i).
%F A306672 a(n) = (L(3*n+2)-1)/2+1.
%F A306672 G.f.: -2*(2*x-1)/((x-1)*(x^2+4*x-1)). - _Alois P. Heinz_, Mar 04 2019
%e A306672 L(0) + L(3) = 6;
%e A306672 L(0) + L(3) + L(6) = 24;
%e A306672 L(0) + L(3) + L(6) + L(9) = 100.
%p A306672 f:= gfun:-rectoproc({a(n + 3) - 5*a(n + 2) + 3*a(n + 1) + a(n), a(0) = 2, a(1) = 6, a(2) = 24},a(n),remember):
%p A306672 map(f, [$0..60]); # _Robert Israel_, Mar 05 2019
%t A306672 Table[(Lucas[3n+2]-1)/2+1,{n,0,25}]
%t A306672 Accumulate[Select[LucasL[Range[0,100]],EvenQ]] (* or *) LinearRecurrence[ {5,-3,-1},{2,6,24},30] (* _Harvey P. Dale_, Jan 18 2021 *)
%o A306672 (PARI) L(n) = fibonacci(n+1)+fibonacci(n-1);
%o A306672 a(n) = sum(k=0, n, L(3*k)); \\ _Michel Marcus_, Mar 05 2019
%o A306672 (Perl) use ntheory ":all"; sub a { vecsum(map{lucasv(1,-1,3*$_)}0..$_[0]) } # _Dana Jacobsen_, Mar 05 2019
%Y A306672 Cf. A000032, A014448, A099919.
%K A306672 nonn
%O A306672 0,1
%A A306672 _Rigoberto Florez_, Mar 04 2019