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A167167 A001045 with a(0) replaced by -1.

%I A167167 #22 Jul 23 2025 20:11:12
%S A167167 -1,1,1,3,5,11,21,43,85,171,341,683,1365,2731,5461,10923,21845,43691,
%T A167167 87381,174763,349525,699051,1398101,2796203,5592405,11184811,22369621,
%U A167167 44739243,89478485,178956971,357913941,715827883,1431655765,2863311531
%N A167167 A001045 with a(0) replaced by -1.
%C A167167 Essentially the same as A001045, and perhaps also A152046.
%C A167167 Also the binomial transform of the sequence with terms (-1)^(n+1)*A128209(n).
%H A167167 G. C. Greubel, <a href="/A167167/b167167.txt">Table of n, a(n) for n = 0..1000</a>
%H A167167 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1,2).
%F A167167 a(n) = A001045(n), n>0.
%F A167167 a(n) + a(n+1) = 2*A001782(n) = 2*A131577(n) = A155559(n) = A090129(n+2), n>0.
%F A167167 G.f.: (2*x^2 + 2*x - 1)/((1+x)*(1-2*x)).
%F A167167 E.g.f.: (exp(2*x) - exp(-x) - 3)/3. - _G. C. Greubel_, Dec 01 2019
%p A167167 seq( `if`(n=0, -1, (2^n -(-1)^n)/3), n=0..35); # _G. C. Greubel_, Dec 01 2019
%t A167167 CoefficientList[Series[(2*x-1+2*x^2)/((1+x)*(1-2*x)), {x, 0, 35}], x] (* _G. C. Greubel_, Jun 04 2016 *)
%t A167167 Table[If[n==0, -1, (2^n -(-1)^n)/3], {n,0,35}] (* _G. C. Greubel_, Dec 01 2019 *)
%t A167167 LinearRecurrence[{1,2},{-1,1,1},40] (* _Harvey P. Dale_, Jul 23 2025 *)
%o A167167 (PARI) vector(36, n, if(n==1, -1, (2^(n-1) +(-1)^n)/3 ) ) \\ _G. C. Greubel_, Dec 01 2019
%o A167167 (Magma) [-1] cat [(2^n -(-1)^n)/3 : n in [1..35]]; // _G. C. Greubel_, Dec 01 2019
%o A167167 (Sage) [-1]+[lucas_number1(n, 1, -2) for n in (1..35)] # _G. C. Greubel_, Dec 01 2019
%o A167167 (GAP) Concatenation([-1], List([1..35], n-> (2^n -(-1)^n)/3) ); # _G. C. Greubel_, Dec 01 2019
%K A167167 sign,less
%O A167167 0,4
%A A167167 _Paul Curtz_, Oct 29 2009
%E A167167 Edited and extended by _R. J. Mathar_, Nov 01 2009