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A086953 Binomial transform of (-1)^mod(n,3) (A257075).

%I A086953 #37 Aug 08 2017 22:14:43
%S A086953 1,0,0,2,6,12,22,42,84,170,342,684,1366,2730,5460,10922,21846,43692,
%T A086953 87382,174762,349524,699050,1398102,2796204,5592406,11184810,22369620,
%U A086953 44739242,89478486,178956972,357913942,715827882,1431655764,2863311530,5726623062
%N A086953 Binomial transform of (-1)^mod(n,3) (A257075).
%H A086953 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).
%F A086953 a(n+3)/2 = A024495(n+2). - corrected by _Vladimir Shevelev_, Aug 08 2017
%F A086953 a(n) = 0^n + Sum{k=0..floor((n-1)/3)} C(n-1, 3*k+2).
%F A086953 a(n) = Sum{k=0..n} C(n, k)(-1)^mod(k, 3).
%F A086953 G.f.: (1 - 3*x + 3*x^2)/((1 - 2*x)*(1 - x + x^2)). - _Paul Barry_, Dec 14 2004
%F A086953 From _Vladimir Shevelev_, Aug 02 2017: (Start)
%F A086953 a(n) = A024493(n) - A131708(n) + A024495(n);
%F A086953 a(n) = A024495(n) if and only if n == 1 (mod 3);
%F A086953 a(n) = A024495(n) - 1 if and only if n == 2 or 3 (mod 6);
%F A086953 a(n) = A024495(n) + 1 if and only if n == 0 or 5 (mod 6);
%F A086953 a(3*k+1) = 2*A024495(3*k). (End)
%F A086953 a(n) = A131370(n+1)/2. - _Rick L. Shepherd_, Aug 02 2017
%F A086953 3*a(n) = 2^n + 2*A057079(n+2). - _R. J. Mathar_, Aug 04 2017
%t A086953 Join[{1, a = 0, b = 0}, Table[c = 2^n - a + b; a = b; b = c, {n, 1, 100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jun 28 2011 *)
%t A086953 LinearRecurrence[{3,-3,2},{1,0,0},40] (* _Harvey P. Dale_, Aug 02 2017 *)
%Y A086953 Cf. A024493, A024495, A131370, A131708, A257075.
%K A086953 nonn,easy
%O A086953 0,4
%A A086953 _Paul Barry_, Jul 25 2003