A111567 Binomial transform of A048654: generalized Pellian with second term equal to 4.

1, 5, 18, 62, 212, 724, 2472, 8440, 28816, 98384, 335904, 1146848, 3915584, 13368640, 45643392, 155836288, 532058368, 1816560896, 6202126848, 21175385600, 72297288704, 246838383616, 842758957056, 2877359060992

Offset

0,2

Comments

Dropping the leading 1, this becomes the 4th row of the 2-shuffle Phi_2(W(sqrt(2)) of the Fraenkel-Kimberling publication. - R. J. Mathar, Aug 17 2009

Floretion Algebra Multiplication Program, FAMP Code: 1lesseq[K*J] with K = + .5'i + .5'j + .5k' + .5'kk' and J = + .5i' + .5j' + 2'kk' + .5'ki' + .5'kj'.

Links

Harvey P. Dale, Table of n, a(n) for n = 0..1000

A. S. Fraenkel and Clark Kimberling, Generalized Wythoff arrays, shuffles and interspersions, Discr. Math. 126 (1-3) (1994) 137-149. [From R. J. Mathar, Aug 17 2009]

Index entries for linear recurrences with constant coefficients, signature (4,-2).

Formula

a(n) = 4*a(n-1) - 2*a(n-2), a(0) = 1, a(1) = 5. Program "FAMP" returns: A111566(n) = A007052(n) - A006012(n) + a(n).

From R. J. Mathar, Apr 02 2008: (Start)

O.g.f.: (1+x)/(1-4*x+2*x^2).

a(n) = A007070(n) + A007070(n-1). (End)

a(n) = ((2+sqrt(18))*(2+sqrt(2))^n) + (2-sqrt(18))*(2-sqrt(2))^n)/4, offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 08 2009

a(n) = ((5+sqrt(32))(2+sqrt(2))^n+(5-sqrt(32))(2-sqrt(2))^n)/2 offset 0. - Al Hakanson (hawkuu(AT)gmail.com), Aug 15 2009

Mathematica

LinearRecurrence[{4, -2}, {1, 5}, 30] (* Harvey P. Dale, Jul 01 2016 *)

Prog

(Maxima)
a[0]:1$
a[1]:5$
a[n]:=4*a[n-1]-2*a[n-2]$
A111567(n):=a[n]$
makelist(A111567(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */

Crossrefs

Cf. A007052, A006012, A111566.

Sequence in context: A255837 A122234 A113301 * A372420 A121050 A029869

Adjacent sequences: A111564 A111565 A111566 * A111568 A111569 A111570

Keyword

easy,nonn

Author

Creighton Dement, Aug 06 2005

Extensions

Typo in definition corrected by Klaus Brockhaus, Aug 09 2009

Status

approved