%I #12 Feb 07 2022 00:05:48
%S 1,3,2,6,6,2,14,24,12,50,66,28,144,216,100,460,648,288,1396,2028,920,
%T 4344,6216,2792,13352,19248,8688,41288,59304,26704,127296,183168,
%U 82576,393040,565056,254592,1212688,1744176,786080
%N Let M = the 3 X 3 matrix [1 1 1; 3 1 0; 2 0 0]. Perform M^n * [1 0 0] getting (1, 3, 2; 6, 6, 2; 14, 24, 12; 50, 66, 28; ...) which we string together to form the sequence.
%C Sequence relating to finite differences.
%C Taking subsets (k = 1,2,3, ...) of three terms: [1, 3, 2; 6, 6, 2; 14, 24, 12; ...), 3 terms in the k-th subset are coefficients in a second degree equation f(x) such that the binomial transform of (k+1)-th subset = terms generated by f(x) of k-th subset. Example: Binomial transform of [14, 24, 12] = 14, 38, 74, 122, ...; f(x)= 6x^2 + 6x + 2. [14, 24, 12] = the 3rd subset of 3 terms, [6, 6, 2] = the second subset. Then, binomial transform of [6, 6, 2] = [6, 12, 20, 33, 42...] such that f(x) = x^2 + 3x + 2, where [1, 3, 2] is the second three term subset of A107271.
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,2,0,0,4,0,0,-2).
%F G.f.: -x*(2*x^6+2*x^5-4*x^3-2*x^2-3*x-1) / (2*x^9-4*x^6-2*x^3+1). [_Colin Barker_, Dec 13 2012]
%e M^3 * [1 0 0] = [14, 24, 12].
%t LinearRecurrence[{0,0,2,0,0,4,0,0,-2},{1,3,2,6,6,2,14,24,12},40] (* _Harvey P. Dale_, Jul 19 2019 *)
%K nonn,easy
%O 1,2
%A _Gary W. Adamson_, May 15 2005
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