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A095128
a(n+3) = 3*a(n+2) + 2*a(n+1) - a(n).
4
1, 4, 13, 46, 160, 559, 1951, 6811, 23776, 82999, 289738, 1011436, 3530785, 12325489, 43026601, 150199996, 524327701, 1830356494, 6389524888, 22304959951, 77863573135, 271811114419, 948855529576, 3312325244431, 11562875678026, 40364421993364, 140906692091713
OFFSET
1,2
COMMENTS
A sequence generated from an inverse Bell matrix, M.
a(n)/a(n-1) tends to 3.4908636153..., which is a root of x^3 - 3*x^2 - 2*x + 1 and an eigenvalue of M. A095127 is generated from the reflected polynomial: x^3 - 2*x^2 - 3*x + 1 and the inverse matrix of M. Bell numbers are sums of row terms of the 3rd-order Stirling number of the second kind matrix shown on p. 171 of Aldrovandi, the matrix being [1 0 0 / 1 1 0 / 1 3 1]. Rotations, or inverses, or related polynomials generate A095125, A095126, A095127, A095128.
REFERENCES
R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, section 13.3.1, "Inverting Bell Matrices", p. 171.
FORMULA
Invert the matrix used to generate A095127, getting M = [3 2 -1 / 1 0 0 / 0 1 0]. Then M^n * [1 1 1] = [p q r] where a(n) = the center term q.
G.f.: (-x^2+x+1)/(x^3-2*x^2-3*x+1). - Harvey P. Dale, Dec 14 2012
EXAMPLE
a(6) = 559 = 3*a(5) + 2*a(4) - a(3) = 3*160 + 2*46 - 13.
a(4) = 46 since M^4 * [1 1 1] = [160 46 13] = [a(5) a(4) a(3)].
MATHEMATICA
a[n_] := (MatrixPower[{{3, 2, -1}, {1, 0, 0}, {0, 1, 0}}, n].{{1}, {1}, {1}})[[2, 1]]; Table[ a[n], {n, 24}] (* Robert G. Wilson v, Jun 01 2004 *)
LinearRecurrence[{3, 2, -1}, {1, 4, 13}, 30] (* Harvey P. Dale, Dec 14 2012 *)
PROG
(Magma) I:=[1, 4, 13]; [n le 3 select I[n] else 3*Self(n-1)+2*Self(n-2)-Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jul 25 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, May 29 2004
EXTENSIONS
Edited and extended by Robert G. Wilson v, Jun 01 2004
a(25)-a(27) from Vincenzo Librandi, Jul 25 2015
STATUS
approved