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A191370 a(n) = 2*(1+(-1)^n)/3 + 2*A010892(n-1).

%I A191370 #28 Sep 06 2022 15:58:41
%S A191370 1,2,4,2,4,8,22,44,88,170,340,680,1366,2732,5464,10922,21844,43688,
%T A191370 87382,174764,349528,699050,1398100,2796200,5592406,11184812,22369624,
%U A191370 44739242
%N A191370 a(n) = 2*(1+(-1)^n)/3 + 2*A010892(n-1).
%C A191370 a(n) and successive differences define an infinite array:
%C A191370     1,   2,   4,   2,   4,   8, ...
%C A191370     1,   2,  -2,   2,   4,  14, ...
%C A191370     1,  -4,   4,   2,  10,   8, ...
%C A191370    -5,   8,  -2,   8,  -2,  14, ...
%C A191370    13, -10,  10, -10,  16,   2, ...
%C A191370   -23,  20, -20,  26, -14,  32, ...
%C A191370   ...
%C A191370 Its main diagonal consists of the powers 2^n. The first upper diagonal is a signed sequence of 2's. The second upper diagonal contains essentially A135440.
%H A191370 Muniru A Asiru, <a href="/A191370/b191370.txt">Table of n, a(n) for n = 0..3000</a>
%H A191370 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1,2).
%F A191370 a(n+3) = 3*2^n - a(n), n >= 0.
%F A191370 a(n+1) - 2*a(n) = -6*A131531(n+1).
%F A191370 a(3*n) = A007613(n), a(1+3*n) = 2*A007613(n), a(2+3*n) = 4*A007613(n).
%F A191370 a(n+6) = a(n) + 21*2^n.
%F A191370 a(n) = ((2^n + 2*(-1)^n)*2^n - 2*i*sqrt(3)*((1+i*sqrt(3))^n - (1-i*sqrt(3))^n))/(3*2^n), where i=sqrt(-1); a(n+1) = 2*(A001045(n) + A010892(n)). - _Bruno Berselli_, Jun 06 2011
%F A191370 G.f.: ( -1+5*x^3 ) / ( (2*x-1)*(1+x)*(x^2-x+1) ). - _R. J. Mathar_, Jun 06 2011
%F A191370 a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4). - _Paul Curtz_, Jun 07 2011
%F A191370 a(n) = A113405(n+3) - 5*A113405(n). - _R. J. Mathar_, Jun 24 2011
%p A191370 A010892 := proc(n) op( 1+(n mod 6),[1,1,0,-1,-1,0]) ; end proc:
%p A191370 A191370 := proc(n) 2^n/3+2*(-1)^n/3+2*A010892(n-1) ; end proc:
%p A191370 seq(A191370(n),n=0..30) ; # _R. J. Mathar_, Jun 06 2011
%t A191370 LinearRecurrence[{2,0,-1,2},{1,2,4,2},30] (* _Harvey P. Dale_, Sep 06 2022 *)
%Y A191370 Cf. A010892, A131531, A007613, A001045.
%Y A191370 Cf. A007283, A024495, A113405.
%K A191370 nonn,easy
%O A191370 0,2
%A A191370 _Paul Curtz_, Jun 01 2011