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%I #22 Mar 11 2024 08:30:22
%S 1,5,18,62,212,724,2472,8440,28816,98384,335904,1146848,3915584,
%T 13368640,45643392,155836288,532058368,1816560896,6202126848,
%U 21175385600,72297288704,246838383616,842758957056,2877359060992
%N Binomial transform of A048654: generalized Pellian with second term equal to 4.
%C 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
%C 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'.
%H Harvey P. Dale, <a href="/A111567/b111567.txt">Table of n, a(n) for n = 0..1000</a>
%H A. S. Fraenkel and Clark Kimberling, <a href="http://dx.doi.org/10.1016/0012-365X(94)90259-3">Generalized Wythoff arrays, shuffles and interspersions</a>, Discr. Math. 126 (1-3) (1994) 137-149. [From _R. J. Mathar_, Aug 17 2009]
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F 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).
%F From _R. J. Mathar_, Apr 02 2008: (Start)
%F O.g.f.: (1+x)/(1-4*x+2*x^2).
%F a(n) = A007070(n) + A007070(n-1). (End)
%F 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
%F 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
%t LinearRecurrence[{4,-2},{1,5},30] (* _Harvey P. Dale_, Jul 01 2016 *)
%o (Maxima)
%o a[0]:1$
%o a[1]:5$
%o a[n]:=4*a[n-1]-2*a[n-2]$
%o A111567(n):=a[n]$
%o makelist(A111567(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%Y Cf. A007052, A006012, A111566.
%K easy,nonn
%O 0,2
%A _Creighton Dement_, Aug 06 2005
%E Typo in definition corrected by _Klaus Brockhaus_, Aug 09 2009
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