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A084170 a(n) = (5*2^n + (-1)^n - 3)/3.

%I A084170 #63 Oct 12 2022 05:23:08
%S A084170 1,2,6,12,26,52,106,212,426,852,1706,3412,6826,13652,27306,54612,
%T A084170 109226,218452,436906,873812,1747626,3495252,6990506,13981012,
%U A084170 27962026,55924052,111848106,223696212,447392426,894784852,1789569706,3579139412
%N A084170 a(n) = (5*2^n + (-1)^n - 3)/3.
%C A084170 Original name of this sequence: Generalized Jacobsthal numbers.
%H A084170 Vincenzo Librandi, <a href="/A084170/b084170.txt">Table of n, a(n) for n = 0..1000</a>
%H A084170 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F A084170 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
%F A084170 a(n) = a(n-1) + 2*a(n-2) + 2, a(0)=1, a(1)=2.
%F A084170 G.f.: (1+x^2)/((1+x)*(1-x)*(1-2*x)).
%F A084170 E.g.f.: 5*exp(2*x)/3 - exp(x) + exp(-x)/3.
%F A084170 a(n+1) = A000975(n+2) + A000975(n).
%F A084170 a(2*n+1) - 2 = 10*A000975(n).
%F A084170 a(2*n+2) - 6 = 20*A000975(n).
%F A084170 a(n+2*k) - a(n) = 5*A002450(k)*2^n = A146882(k-1)*2^n, k >= 0. - _Paul Curtz_, Jun 15 2011
%F A084170 From _Yosu Yurramendi_, Jul 05 2016: (Start)
%F A084170 a(n) = A169969(2n) - 1, n >= 1; a(n) = 3*2^(n-1) - 1 + A169969(2n-7), n >= 5.
%F A084170 a(n+3) = 15*2^n - 2 - a(n), n >= 0, a(0)=1, a(1)=2, a(2)=6.
%F A084170 a(n) + A026644(n) = 3*2^n - 2, n >= 1.
%F A084170 a(n+3) = 3*2^(n+2) + A026644(n), n >= 1. (End)
%F A084170 a(n) = A000225(n+1) - A001045(n). - _Yuchun Ji_, Mar 17 2020
%t A084170 LinearRecurrence[{2,1,-2},{1,2,6},40] (* or *) Table[(5*2^n+(-1)^n-3)/3,{n,0,40}] (* _Harvey P. Dale_, Jan 29 2012 *)
%o A084170 (PARI) a(n)=(5*2^n)\/3-1 \\ _Charles R Greathouse IV_, Jul 01 2011
%o A084170 (Magma) [(5*2^n +(-1)^n)/3 -1: n in [0..35]]; // _Vincenzo Librandi_, Jul 05 2011
%o A084170 (SageMath) [(2/3)*(5*2^(n-1) -1 -(n%2)) for n in range(41)] # _G. C. Greubel_, Oct 11 2022
%Y A084170 Cf. A000975, A002450, A026644, A146882, A169969.
%Y A084170 Cf. A000225 (Mersenne numbers), A001045 (Jacobsthal numbers).
%K A084170 easy,nonn
%O A084170 0,2
%A A084170 _Paul Barry_, May 18 2003