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

%I #32 Feb 07 2021 17:10:13

%S 0,1,2,5,13,30,63,127,254,509,1021,2046,4095,8191,16382,32765,65533,

%T 131070,262143,524287,1048574,2097149,4194301,8388606,16777215,

%U 33554431,67108862,134217725,268435453,536870910,1073741823,2147483647,4294967294,8589934589

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

%C After 0, partial sums of A281166.

%C Table of the first differences:

%C 0, 1, 2, 5, 13, 30, 63, 127, 254, 509, 1021, 2046, ...

%C 1, 1, 3, 8, 17, 33, 64, 127, 255, 512, 1025, 2049, ... A281166

%C 0, 2, 5, 9, 16, 31, 63, 128, 257, 513, 1024, 2047, ...

%C 2, 3, 4, 7, 15, 32, 65, 129, 256, 511, 1023, 2048, ...

%C repeat A281166.

%H Colin Barker, <a href="/A282153/b282153.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,5,-2).

%F From _Colin Barker_, Feb 10 2017: (Start)

%F G.f.: x*(1 - 2*x + 3*x^2)/((1 - x)*(1 - 2*x)*(1 - x + x^2)).

%F a(n) = 4*a(n-1) - 6*a(n-2) + 5*a(n-3) - 2*a(n-4) for n>3. (End)

%F From _Bruno Berselli_, Feb 10 2017: (Start)

%F a(n) = 2^n + ((-1)^floor(n/3) + (-1)^floor((n+1)/3))/2 - 2. Therefore:

%F a(3*k) = 8^k + (-1)^k - 2,

%F a(3*k+1) = 2*8^k + (-1)^k - 2,

%F a(3*k+2) = 4*8^k - 2. (End)

%F 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 a(n) - (a(n) mod 9) = A153237(n) = 9*A153234(n).

%t LinearRecurrence[{4, -6, 5, -2}, {0, 1, 2, 5}, 34] (* _Robert P. P. McKone_, Feb 07 2021 *)

%o (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 Cf. A000079, A153237, A281166.

%K nonn,easy

%O 0,3

%A _Paul Curtz_, Feb 07 2017

%E More terms from _Colin Barker_, Feb 10 2017