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

Author

Creighton Dement, Aug 06 2005

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A007052, A006012, A111566.

Programs

Mathematica

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

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 */

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

Extensions

Typo in definition corrected by Klaus Brockhaus, Aug 09 2009