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A141775 Binomial transform of (1, 2, 0, 1, 2, 0, 1, 2, 0, ...).

%I A141775 #25 Sep 08 2022 08:45:35
%S A141775 1,3,5,8,15,31,64,129,257,512,1023,2047,4096,8193,16385,32768,65535,
%T A141775 131071,262144,524289,1048577,2097152,4194303,8388607,16777216,
%U A141775 33554433,67108865,134217728,268435455,536870911,1073741824,2147483649,4294967297,8589934592,17179869183
%N A141775 Binomial transform of (1, 2, 0, 1, 2, 0, 1, 2, 0, ...).
%C A141775 From _Paul Curtz_, Jun 15 2011: (Start)
%C A141775 A square array of a(n) and its higher order differences is defined by T(0,k) = a(k) and T(n,k) = T(n-1,k+1)-T(n-1,k):
%C A141775 1, 3, 5, 8, 15, 31,
%C A141775 2, 2, 3, 7, 16, 33,
%C A141775 0, 1, 4, 9, 17, 32, see A130785(n).
%C A141775 1, 3, 5, 8, 15, 31,
%C A141775 2, 2, 3, 7, 16, 33,
%C A141775 a(n) is identical to its third differences: T(n+3,k) = T(n,k).
%C A141775 The main diagonal is T(n,n) = 2^n. Subdiagonals are T(n,n-1) = A014551(n) and T(n,n-2) = A062510(n).
%C A141775 (End)
%H A141775 Harvey P. Dale, <a href="/A141775/b141775.txt">Table of n, a(n) for n = 0..1000</a>
%H A141775 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2).
%F A141775 From _Paul Curtz_, Jun 15 2011: (Start)
%F A141775 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3).
%F A141775 a(n) = 2^n - A128834(n).
%F A141775 a(n) - 2a(n-1)= A057079(n+1).
%F A141775 a(n) + a(n+3) = 9*2^n.
%F A141775 a(n+6) - a(n) = 63*2^n.
%F A141775 a(n) = A130785(n) - A130785(n-1). (End)
%F A141775 G.f.: (x-1)*(1+x) / ( (2*x-1)*(x^2-x+1) ). - _R. J. Mathar_, Jun 22 2011
%F A141775 a(n) = 2^n + (2*sin((Pi*n)/3))/sqrt(3). - _Colin Barker_, Feb 10 2017
%e A141775 a(4) = 8 = (1, 2, 0, 1) dot (1, 3, 3, 1) = (1 + 6 + 0 + 1).
%t A141775 LinearRecurrence[{3,-3,2},{1,3,5},40] (* _Harvey P. Dale_, May 29 2012 *)
%o A141775 (PARI) x='x+O('x^30); Vec((x-1)*(1+x)/((2*x-1)*(x^2-x+1))) \\ _G. C. Greubel_, Jan 15 2018
%o A141775 (Magma) I:=[1,3,5]; [n le 3 select I[n] else 3*Self(n-1) - 3*Self(n-2) + 2*Self(n-3): n in [1..30]]; // _G. C. Greubel_, Jan 15 2018
%Y A141775 Cf. A000079, A057079.
%K A141775 nonn,easy
%O A141775 0,2
%A A141775 _Gary W. Adamson_, Jul 03 2008