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A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).

%I A167784 #44 Sep 05 2025 10:24:08
%S A167784 1,0,4,2,16,14,64,74,256,350,1024,1562,4096,6734,16384,28394,65536,
%T A167784 117950,262144,484922,1048576,1979054,4194304,8034314,16777216,
%U A167784 32491550,67108864,131029082,268435456,527304974,1073741824,2118785834,4294967296,8503841150
%N A167784 a(n) = 2^n - (1 - (-1)^n)*3^((n-1)/2).
%C A167784 Binomial transform of A077917, the signed variant of A127864.
%H A167784 Robert Israel, <a href="/A167784/b167784.txt">Table of n, a(n) for n = 0..3318</a>
%H A167784 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,3,-6).
%F A167784 a(n) = A167936(n+1) - A167936(n).
%F A167784 a(2n) = A000302(n). a(2n+1) = 2*A005061(n).
%F A167784 a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3).
%F A167784 G.f.: (x-1)^2/((2*x-1)*(3*x^2-1)).
%F A167784 a(n+4) mod 9 = A153130(n+4) = A146501(n+2), n>=0.
%F A167784 E.g.f.: exp(2*x) - (2/sqrt(3))*sinh(sqrt(3)*x). - _G. C. Greubel_, Jun 27 2016
%p A167784 seq(2^n - (1 - (-1)^n)*3^((n-1)/2), n=0..100); # _Robert Israel_, Apr 11 2019
%t A167784 LinearRecurrence[{2, 3, -6}, {1, 0, 4}, 40] (* _Harvey P. Dale_, Nov 29 2011 *)
%Y A167784 Cf. A000302, A005061, A077917, A127864, A146501, A153130, A167936.
%Y A167784 Cf. A154383.
%K A167784 nonn,easy,changed
%O A167784 0,3
%A A167784 _Paul Curtz_, Nov 12 2009
%E A167784 Edited and extended by _R. J. Mathar_, Feb 27 2010
%E A167784 Incorrect b-file corrected by _Robert Israel_, Apr 11 2019