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A282153 Expansion of x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).

%I A282153 #32 Feb 07 2021 17:10:13
%S A282153 0,1,2,5,13,30,63,127,254,509,1021,2046,4095,8191,16382,32765,65533,
%T A282153 131070,262143,524287,1048574,2097149,4194301,8388606,16777215,
%U A282153 33554431,67108862,134217725,268435453,536870910,1073741823,2147483647,4294967294,8589934589
%N A282153 Expansion of x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
%C A282153 After 0, partial sums of A281166.
%C A282153 Table of the first differences:
%C A282153 0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, ...
%C A282153 1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, ... A281166
%C A282153 0, 2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, ...
%C A282153 2, 3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, ...
%C A282153 repeat A281166.
%H A282153 Colin Barker, <a href="/A282153/b282153.txt">Table of n, a(n) for n = 0..1000</a>
%H A282153 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-2).
%F A282153 From _Colin Barker_, Feb 10 2017: (Start)
%F A282153 G.f.: x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).
%F A282153 a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n>3. (End)
%F A282153 From _Bruno Berselli_, Feb 10 2017: (Start)
%F A282153 a(n) = 2^n + ((-1)^floor(n/3) + (-1)^floor((n+1)/3))/2 - 2. Therefore:
%F A282153 a(3*k)   =   8^k + (-1)^k - 2,
%F A282153 a(3*k+1) = 2*8^k + (-1)^k - 2,
%F A282153 a(3*k+2) = 4*8^k - 2. (End)
%F A282153 a(n+6*h) = a(n) + 2^n*(64^h - 1) with h>=0. For h=1, a(n+6) = a(n) + 63*2^n.
%F A282153 a(n) - (a(n) mod 9) = A153237(n) = 9*A153234(n).
%t A282153 LinearRecurrence[{4, -6, 5, -2}, {0, 1, 2, 5}, 34] (* _Robert P. P. McKone_, Feb 07 2021 *)
%o A282153 (PARI) concat(0, Vec(x*(1 - 2*x + 3*x^2) / ((1 - x)*(1 - 2*x)*(1 - x + x^2)) + O(x^50))) \\ _Colin Barker_, Feb 10 2017
%Y A282153 Cf. A000079, A153237, A281166.
%K A282153 nonn,easy
%O A282153 0,3
%A A282153 _Paul Curtz_, Feb 07 2017
%E A282153 More terms from _Colin Barker_, Feb 10 2017