|
|
A095262
|
|
A sequence derived from a truncated Pascal's Triangle matrix.
|
|
1
|
|
|
2, 21, 137, 735, 3557, 16191, 70877, 302295, 1266437, 5239311, 21481517, 87506055, 354778517, 1433405631, 5776554557, 23235129015, 93327477797, 374471255151, 1501369969997, 6015936563175, 24095119972277, 96474608387871
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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)].
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
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|