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A153643 Jacobsthal numbers A001045 incremented by 2.

%I A153643 #27 Apr 18 2025 22:22:43
%S A153643 2,3,3,5,7,13,23,45,87,173,343,685,1367,2733,5463,10925,21847,43693,
%T A153643 87383,174765,349527,699053,1398103,2796205,5592407,11184813,22369623,
%U A153643 44739245,89478487,178956973,357913943,715827885,1431655767,2863311533,5726623063
%N A153643 Jacobsthal numbers A001045 incremented by 2.
%H A153643 G. C. Greubel, <a href="/A153643/b153643.txt">Table of n, a(n) for n = 0..1000</a>
%H A153643 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F A153643 a(n) = 2 + A001045(n) = A001045(n) + A007395(n) = 1 + A128209(n).
%F A153643 a(n) - A010684(n) = A078008(n), first differences of A001045. - _Paul Curtz_, Jan 17 2009
%F A153643 G.f.: (2 - x - 5*x^2)/((1+x)*(1-x)*(1-2*x)). - _R. J. Mathar_, Jan 23 2009
%F A153643 a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) for n >= 3. - _Andrew Howroyd_, Feb 26 2018
%t A153643 LinearRecurrence[{1,2},{0,1}, 40] + 2 (* _Harvey P. Dale_, May 26 2014 *)
%t A153643 LinearRecurrence[{2,1,-2},{2,3,3}, 40] (* _Georg Fischer_, Apr 02 2019 *)
%o A153643 (PARI) my(x='x+O('x^40)); Vec( (2-x-5*x^2)/((1-x^2)*(1-2*x)) ) \\ _G. C. Greubel_, Apr 02 2019
%o A153643 (Magma) I:=[2,3,3]; [n le 3 select I[n] else 2*Self(n-1) +Self(n-2) -2*Self(n-3): n in [1..40]]; // _G. C. Greubel_, Apr 02 2019
%o A153643 (Sage) ((2-x-5*x^2)/((1-x^2)*(1-2*x))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Apr 02 2019
%o A153643 (GAP) a:=[2,3,3];; for n in [4..40] do a[n]:=2*a[n-1]+a[n-2]-2*a[n-3]; od; a; # _G. C. Greubel_, Apr 02 2019
%o A153643 (Python)
%o A153643 def A153643(n): return ((1<<n)|1)//3+2 # _Chai Wah Wu_, Apr 18 2025
%Y A153643 Cf. A001045, A007395, A010684, A078008, A128209.
%K A153643 nonn,easy
%O A153643 0,1
%A A153643 _Paul Curtz_, Dec 30 2008
%E A153643 Edited and extended by _R. J. Mathar_, Jan 23 2009