A093406 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) + a(n-4).

1, 3, 11, 31, 71, 145, 289, 601, 1321, 2979, 6683, 14743, 32111, 69697, 151777, 332113, 728689, 1598883, 3503627, 7668079, 16774775, 36704017, 80343361, 175916521, 385196761, 843365379, 1846290395, 4041672871, 8847607391, 19368919297, 42403014721, 92830645537

Offset: 1

Author

Gary W. Adamson, Mar 28 2004

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to 2.189207115... = 1 + 2^(1/4) = 1 + A010767.

Examples

			a(4) = 31, since M^4 * [1,1,1,1] = [3, 11, 31, 71].
		

References

• E. J. Barbeau, Polynomials, Springer-Verlag NY Inc, 1989, p. 136.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,1).

Crossrefs

Cf. A010767, A052101.

Programs

Mathematica

LinearRecurrence[{4,-6,4,1},{1,3,11,31},40] (* Harvey P. Dale, Jul 22 2013 *)

Formula

We use a 4 X 4 matrix corresponding to the characteristic polynomial (x - 1)^4 - 2 = 0 = x^4 - 4x^3 + 6x^2 - 4x - 1 = 0, being [0 1 0 0 / 0 0 1 0 / 0 0 0 1 / 1 4 -6 4]. Let the matrix = M. Perform M^n * [1, 1, 1, 1]. a(n) = the third term from the left, (the other 3 terms being offset members of the series).

a(n) = 4*a(n-1)-6*a(n-2)+4*a(n-3)+a(n-4). G.f.: -x*(x^3+5*x^2-x+1)/ (x^4+4*x^3-6*x^2+4*x-1). [Colin Barker, Oct 21 2012]

Extensions

Corrected by T. D. Noe, Nov 08 2006

New name using recurrence from Colin Barker, Joerg Arndt, Apr 15 2021