%I #16 Jan 05 2025 23:38:26

%S 2,21,137,735,3557,16191,70877,302295,1266437,5239311,21481517,

%T 87506055,354778517,1433405631,5776554557,23235129015,93327477797,

%U 374471255151,1501369969997,6015936563175,24095119972277,96474608387871

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

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (9,-26,24).

%F 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)].

%F From _Colin Barker_, Oct 21 2012: (Start)

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

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

%e 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.

%t 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 *)

%Y Cf. A001047.

%K nonn,easy

%O 1,1

%A _Gary W. Adamson_, May 31 2004

%E Edited, corrected and extended by _Robert G. Wilson v_, Jun 05 2004