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

%I A111566 #41 May 26 2024 08:25:12
%S A111566 1,6,22,76,260,888,3032,10352,35344,120672,412000,1406656,4802624,
%T A111566 16397184,55983488,191139584,652591360,2228086272,7607162368,
%U A111566 25972476928,88675582976,302757378048,1033678346240,3529198628864,12049437822976,41139354034176,140458540490752
%N A111566 a(n) = ((1+sqrt(8))*(2+sqrt(2))^n + (1-sqrt(8))*(2-sqrt(2))^n)/2.
%C A111566 Binomial transform of A048655: generalized Pellian with second term equal to 5.
%C A111566 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'.
%H A111566 G. C. Greubel, <a href="/A111566/b111566.txt">Table of n, a(n) for n = 0..1000</a>
%H A111566 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-2).
%F A111566 a(n) = 4*a(n-1) - 2*a(n-2), a(0) = 1, a(1) = 6.
%F A111566 Program "FAMP" returns: a(n) = A007052(n) - A006012(n) + A111567(n).
%F A111566 From _R. J. Mathar_, Apr 02 2008: (Start)
%F A111566 O.g.f.: (1+2*x)/(1-4*x+2*x^2).
%F A111566 a(n) = A007070(n) + 2*A007070(n-1). (End)
%F A111566 a(n) = Sum_{k=0..n} A207543(n,k)*2^k. - _Philippe Deléham_, Feb 25 2012
%F A111566 a(n) = 4*A007070(n) - A007052(n+1). - _Yuriy Sibirmovsky_, Sep 13 2016
%F A111566 E.g.f.: exp(2*x)*(cosh(sqrt(2)*x) + 2*sqrt(2)*sinh(sqrt(2)*x)). - _Stefano Spezia_, May 26 2024
%t A111566 LinearRecurrence[{4,-2},{1,6},30] (* _Harvey P. Dale_, Jan 31 2015 *)
%o A111566 (Magma) Z<x>:=PolynomialRing(Integers()); N<r>:=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
%o A111566 (PARI) x='x+O('x^30); Vec((1+2*x)/(1-4*x+2*x^2)) \\ _G. C. Greubel_, Jan 27 2018
%Y A111566 Cf. A007052, A006012, A111567, A007070.
%K A111566 easy,nonn
%O A111566 0,2
%A A111566 _Creighton Dement_, Aug 06 2005
%E A111566 Edited by _N. J. A. Sloane_, Jul 27 2009 using new definition from Al Hakanson (hawkuu(AT)gmail.com)