A111566 a(n) = ((1+sqrt(8))*(2+sqrt(2))^n + (1-sqrt(8))*(2-sqrt(2))^n)/2.

1, 6, 22, 76, 260, 888, 3032, 10352, 35344, 120672, 412000, 1406656, 4802624, 16397184, 55983488, 191139584, 652591360, 2228086272, 7607162368, 25972476928, 88675582976, 302757378048, 1033678346240, 3529198628864, 12049437822976, 41139354034176, 140458540490752

Offset: 0

Author

Creighton Dement, Aug 06 2005

Keywords

• easy
• nonn

Comments

Binomial transform of A048655: generalized Pellian with second term equal to 5.

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-2).

Crossrefs

Cf. A007052, A006012, A111567, A007070.

Programs

Magma

Z:=PolynomialRing(Integers()); N:=NumberField(x^2-2); S:=[ ((1+2*r)*(2+r)^n+(1-2*r)*(2-r)^n)/2: n in [0..23] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jul 27 2009

Mathematica

LinearRecurrence[{4,-2},{1,6},30] (* Harvey P. Dale, Jan 31 2015 *)

PARI

x='x+O('x^30); Vec((1+2*x)/(1-4*x+2*x^2)) \\ G. C. Greubel, Jan 27 2018

Formula

a(n) = 4*a(n-1) - 2*a(n-2), a(0) = 1, a(1) = 6.

Program "FAMP" returns: a(n) = A007052(n) - A006012(n) + A111567(n).

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

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

a(n) = A007070(n) + 2*A007070(n-1). (End)

a(n) = Sum_{k=0..n} A207543(n,k)*2^k. - Philippe Deléham, Feb 25 2012

a(n) = 4*A007070(n) - A007052(n+1). - Yuriy Sibirmovsky, Sep 13 2016

E.g.f.: exp(2*x)*(cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x)). - Stefano Spezia, May 26 2024

Extensions

Edited by N. J. A. Sloane, Jul 27 2009 using new definition from Al Hakanson (hawkuu(AT)gmail.com)