A095262 A sequence derived from a truncated Pascal's Triangle matrix.

2, 21, 137, 735, 3557, 16191, 70877, 302295, 1266437, 5239311, 21481517, 87506055, 354778517, 1433405631, 5776554557, 23235129015, 93327477797, 374471255151, 1501369969997, 6015936563175, 24095119972277, 96474608387871

Offset

1,1

Comments

The recursive multipliers (9), (-26), (24) are present with changed signs in the characteristic polynomial of M: x^3 - 9x^2 + 26x - 24.

Links

Table of n, a(n) for n=1..22.

Index entries for linear recurrences with constant coefficients, signature (9,-26,24).

Formula

a(n+3) = 9*a(n+2) - 26*a(n+1) + 24*a(n), a(1) = 2, a(2) = 31, a(3) = 137. Let M = the 3 X 3 matrix [2 0 0 / 3 3 0 / 4 6 4] (derived from Pascal's triangle rows by deleting the 1's and filling in with 0's). Then M^n * [1 0 0] = [2^n 3*A001047(n) 2*A095262(n)].

From Colin Barker, Oct 21 2012: (Start)

a(n) = (7*2^n-2*3^(2+n)+11*4^n)/2.

G.f.: -x*(3*x+2)/((2*x-1)*(3*x-1)*(4*x-1)). (End)

Example

a(5) = 3557 = 9*735 - 26*137 + 24*21. a(4) = 735 since M^4 *[1 0 0] = [2^4 3*A001047(n) 2*A095262(n)] = [16 195 1470]. Then 735 = 1470/2.

Mathematica

a[n_] := (MatrixPower[{{2, 0, 0}, {3, 3, 0}, {4, 6, 4}}, n].{{1}, {0}, {0}})[[3, 1]]/2; Table[ a[n], {n, 22}] (* Robert G. Wilson v, Jun 05 2004 *)

Crossrefs

Cf. A001047.

Sequence in context: A136588 A171009 A098661 * A209519 A377859 A215710

Adjacent sequences: A095259 A095260 A095261 * A095263 A095264 A095265

Keyword

nonn,easy

Author

Gary W. Adamson, May 31 2004

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jun 05 2004

Status

approved