Search: id:A030195
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%I A030195
%S A030195 0,1,3,12,45,171,648,2457,9315,35316,133893,507627,1924560,7296561,
%T A030195 27663363,104879772,397629405,1507527531,5715470808,21668995017,
%U A030195 82153397475,311467177476,1180861724853,4476986706987,16973545295520
%N A030195 a(n) = 3*a(n-1)+3*a(n-2), a(0)=0, a(1)=1.
%C A030195 Scaled Chebyshev U-polynomials evaluated at I*sqrt(3)/2.
%D A030195 A. F. Horadam, Special properties of the sequence w_n(a,b; p,q), Fib. 
               Quart., 5.5 (1967), 424-434. Case n->n+1, a=0,b=1; p=q=3.
%D A030195 Thomas Koshy, "Fibonacci and Lucas Numbers with Applications", Wiley, 
               2001, p. 471.
%D A030195 W. Lang, On polynomials related to powers of the generating function 
               of Catalan's numbers, Fib. Quart. 38,5 (2000) 408-419; Eqs. (39), 
               (41) and (45), rhs, m=3.
%H A030195 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A030195 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/
               RecursiveSequences.html">Recursive Sequences</a>
%H A030195 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A030195 a(n)=(-I*sqrt(3))^n*U(n, I*sqrt(3)/2), g.f.: 1/(1-3*x-3*x^2).
%F A030195 a(n) = sum(3^(n-k)*binomial(n-k, k), k=0..floor(n/2)). - Emeric Deutsch 
               (deutsch(AT)duke.poly.edu), Nov 14 2001
%F A030195 a(n) = [p^(n+1) - q^(n+1)]/(sqrt 21); p = (3 + sqrt 21)/2, q = (3 - sqrt 
               21)/2. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 02 2003
%F A030195 For n > 0, a(n) = Sum_{k=0..n-1} (2^k)*A063967(n-1,k) - Gerald McGarvey 
               (gerald.mcgarvey(AT)comcast.net), Jul 23 2006
%F A030195 a(n+1)=Sum_{k, 0<=k<=n}2^k*A063967(n,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 03 2006
%F A030195 G.f.: x/(1-3x-3x^2). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 19 2008]
%t A030195 CoefficientList[Series[1/(1-3x-3x^2), {x, 0, 25}], x] - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Mar 22 2007
%o A030195 (Other) sage: [lucas_number1(n,3,-3) for n in xrange(0, 25)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%Y A030195 Equals round(A085480(n)/sqrt(21)).
%Y A030195 Sequence in context: A062561 A128593 A085481 this_sequence A114515 A151162 
               A094547
%Y A030195 Adjacent sequences: A030192 A030193 A030194 this_sequence A030196 A030197 
               A030198
%K A030195 nonn
%O A030195 0,3
%A A030195 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)
%E A030195 Edited by Ralf Stephan, Aug 02 2004
%E A030195 I simplified the definition. As a result the offsets in some of the formulae 
               may need to shifted by 1. - N. J. A. Sloane (njas(AT)research.att.com), 
               Apr 01, 2006.

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