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%I #25 May 08 2017 18:43:51
%S 3,7,12,25,48,97,192,385,768,1537,3072,6145,12288,24577,49152,98305,
%T 196608,393217,786432,1572865,3145728,6291457,12582912,25165825,
%U 50331648,100663297,201326592,402653185,805306368,1610612737,3221225472,6442450945,12884901888
%N a(0) = 3, a(n+1) = 2*a(n) + periodic sequence of length 2: repeat [1, -2].
%C a(n) mod 9 is a periodic sequence of length 2: repeat [3, 7].
%C From 7, the last digit is of period 4: repeat [7, 2, 5, 8].
%C (Main sequence for the signature (2,1,-2): 0, 0, 1, 2, 5, 10, 21, 42, ... = 0 followed by A000975(n) = b(n), which first differences are A001045(n) (Paul Barry, Oct 08 2005). Then, 0 followed by b(n) is an autosequence of the first kind. The corresponding autosequence of the second kind is 0, 0, 2, 3, 8, 15, 32, 63, ... . See A277078(n).)
%C Difference table of a(n):
%C 3, 7, 12, 25, 48, 97, 192, ...
%C 4, 5, 13, 23, 49, 95, 193, ... = -(-1)^n* A140683(n)
%C 1, 8, 10, 26, 46, 98, 190, ... = A259713(n)
%C 7, 2, 16, 20, 52, 92, 196, ...
%C -5, 14, 4, 32, 40, 104, 184, ...
%C ... .
%H Colin Barker, <a href="/A280345/b280345.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F a(2n) = 3*4^n, a(2n+1) = 6*4^n + 1.
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3), n>2.
%F a(n+2) = a(n) + 9*2^n.
%F a(n) = 2^(n+2) - A051049(n).
%F From _Colin Barker_, Jan 01 2017: (Start)
%F a(n) = 3*2^n for n even.
%F a(n) = 3*2^n + 1 for n odd.
%F G.f.: (3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)).
%F (End)
%F Binomial transform of 3, followed by (-1)^n* A140657(n).
%e a(0) = 3, a(1) = 2*3 + 1 = 7, a(2) = 2*7 - 2 = 12, a(3) = 2*12 + 1 = 25.
%t a[0] = 3; a[n_] := a[n] = 2 a[n - 1] + 1 + (-3) Boole[EvenQ@ n]; Table[a@ n, {n, 0, 32}] (* or *)
%t CoefficientList[Series[(3 + x - 5 x^2)/((1 - x) (1 + x) (1 - 2 x)), {x, 0, 32}], x] (* _Michael De Vlieger_, Jan 01 2017 *)
%o (PARI) Vec((3 + x - 5*x^2) / ((1 - x)*(1 + x)*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, Jan 01 2017
%Y Cf. A005010, A051049, A140657, A140683, A164346, A199116, A259713.
%K nonn,easy
%O 0,1
%A _Paul Curtz_, Jan 01 2017
%E More terms from _Colin Barker_, Jan 01 2017