A107271 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.

1, 3, 2, 6, 6, 2, 14, 24, 12, 50, 66, 28, 144, 216, 100, 460, 648, 288, 1396, 2028, 920, 4344, 6216, 2792, 13352, 19248, 8688, 41288, 59304, 26704, 127296, 183168, 82576, 393040, 565056, 254592, 1212688, 1744176, 786080

Offset

1,2

Comments

Sequence relating to finite differences.

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.

Links

Table of n, a(n) for n=1..39.

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,4,0,0,-2).

Formula

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]

Example

M^3 * [1 0 0] = [14, 24, 12].

Mathematica

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 *)

Crossrefs

Sequence in context: A306443 A336518 A189073 * A196565 A104633 A102022

Adjacent sequences: A107268 A107269 A107270 * A107272 A107273 A107274

Keyword

nonn,easy

Author

Gary W. Adamson, May 15 2005

Status

approved