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Expansion of 1 / ((1-2*x)*(1-4*x+x^2)).
3

%I #22 Aug 06 2021 05:10:05

%S 1,6,27,110,429,1638,6187,23238,87021,325358,1215435,4538430,16942381,

%T 63239286,236031147,880918070,3287706669,12270039678,45792714187,

%U 170901341358,637813699821,2380355555078,8883612714795,33154103692710,123732818833261,461777205194766,1723376069054667

%N Expansion of 1 / ((1-2*x)*(1-4*x+x^2)).

%H Colin Barker, <a href="/A216263/b216263.txt">Table of n, a(n) for n = 0..1000</a>

%H László Németh and László Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL24/Nemeth/nemeth8.html">Sequences Involving Square Zig-Zag Shapes</a>, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-9,2).

%F G.f.: 1/((1-2*x)*(1-4*x+x^2)).

%F a(n) = 6*a(n-1) - 9*a(n-2) + 2*a(n-3), a(0) = 1, a(1) = 6, a(2) = 27.

%F 3*a(n) = -2^(n+2) + A001075(n+2). - _R. J. Mathar_, Mar 29 2013

%F a(n) = (-2^(3+n) + (7-4*sqrt(3))*(2-sqrt(3))^n + (2+sqrt(3))^n*(7+4*sqrt(3))) / 6. - _Colin Barker_, Feb 05 2017

%t CoefficientList[Series[1/((1 - 2 x)*(1 - 4 x + x^2)), {x, 0, 26}], x] (* _Michael De Vlieger_, Aug 05 2021 *)

%o (PARI) Vec(1/((1-2*x)*(1-4*x+x^2)) + O(x^30)) \\ _Colin Barker_, Feb 05 2017

%Y A diagonal of A214846.

%Y Cf. A001075.

%K nonn,easy

%O 0,2

%A _Philippe Deléham_, Mar 15 2013