|
Search: id:A100058
|
|
|
| A100058 |
|
Expansion of 1 / (1 - 3x - x^2 + 2x^3). |
|
+0
4
|
|
| 1, 3, 10, 31, 97, 302, 941, 2931, 9130, 28439, 88585, 275934, 859509, 2677291, 8339514, 25976815, 80915377, 252043918, 785093501, 2445493667, 7617486666, 23727766663, 73909799321, 230222191294, 717120839877, 2233765112283
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
a(n)/a(n-1) tends to 3.1149075414...an eigenvalue of the matrix M and a root of the characteristic polynomial x^3 - 3x^2 - x + 2.
|
|
REFERENCES
|
Boris A. Bondarenko, "Generalized Pascal Triangles and Pyramids, Their Fractals, Graphs and Applications", Fibonacci Association, 1993, p. 27.
|
|
FORMULA
|
Recurrence: a(0) = 1, a(1) = 3, a(2) = 10; a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
Given Hosoya's triangle: 1; 1, 1; 2, 1, 2; considered as an upper triangular 3 X 3 matrix M: [2 1 2 / 1 1 0 / 1 0 0]; a(n) = center term in M^n * [1 0 0].
|
|
EXAMPLE
|
a(5) = 97, center term in M^5 * [1 0 0]: [205 97 66].
|
|
MATHEMATICA
|
CoefficientList[Series[1/(1 - 3x - x^2 + 2x^3), {x, 0, 25}], x] (* Or *)
Table[(MatrixPower[{{2, 1, 2}, {1, 1, 0}, {1, 0, 0}}, n].{1, 0, 0})[[2]], {n, 26}] (from Robert G. Wilson v Nov 04 2004)
|
|
CROSSREFS
|
Partial sums of A052911. Cf. A019481, A052550, A052939, A100059, A058071.
Sequence in context: A055217 A097472 A068094 this_sequence A002160 A114487 A017934
Adjacent sequences: A100055 A100056 A100057 this_sequence A100059 A100060 A100061
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 31 2004
|
|
EXTENSIONS
|
Edited by Ralf Stephan, Nov 02 2004
Corrected and extended by Robert G. Wilson v (rgwv(AT)rgwv.com), Nov 04 2004
|
|
|
Search completed in 0.002 seconds
|
