A095128 a(n+3) = 3*a(n+2) + 2*a(n+1) - a(n).

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

Author

Gary W. Adamson, May 29 2004

Keywords

• nonn

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.

Examples

			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)].
		

References

• R. Aldrovandi, "Special Matrices of Mathematical Physics", World Scientific, 2001, section 13.3.1, "Inverting Bell Matrices", p. 171.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,2,-1).

Crossrefs

Cf. A095125, A095126, A095127.

Programs

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

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 *)

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

Extensions

Edited and extended by Robert G. Wilson v, Jun 01 2004

a(25)-a(27) from Vincenzo Librandi, Jul 25 2015