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

1,2

Comments

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

References

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

Links

Table of n, a(n) for n=1..32.

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

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]

Example

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

Mathematica

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

Crossrefs

Cf. A010767, A052101.

Sequence in context: A261148 A071568 A097081 * A107587 A245931 A190590

Adjacent sequences: A093403 A093404 A093405 * A093407 A093408 A093409

Keyword

nonn,easy

Author

Gary W. Adamson, Mar 28 2004

Extensions

Corrected by T. D. Noe, Nov 08 2006

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

Status

approved