%I A084057
%S A084057 1,1,6,16,56,176,576,1856,6016,19456,62976,203776,659456,2134016,
%T A084057 6905856,22347776,72318976,234029056,757334016,2450784256,7930904576,
%U A084057 25664946176,83053510656,268766806016,869747654656,2814562533376
%N A084057 a(n)=2a(n-1)+4a(n-2), a(0)=1, a(1)=1.
%C A084057 Inverse binomial transform of A001077. Binomial transform of expansion 
               of cosh(sqrt(5)x) (1,0,5,0,25,...).
%C A084057 The same sequence may be obtained by the following process. Starting 
               a priori with the fraction 1/1, the numerators of fractions built 
               according to the rule: add top and bottom to get the new bottom, 
               add top and 5 times the bottom to get the new top. The limit of the 
               sequence of fractions is sqrt(5). - Cino Hilliard (hillcino368(AT)gmail.com), 
               Sep 25 2005
%C A084057 Equals right border of triangle A143969. (1, 6, 16, 56,...) = row sums 
               of triangle A143969 and INVERT transform of (1, 5, 5, 5,...). [From 
               Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008]
%D A084057 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%H A084057 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A084057 a(n)=((1+sqrt(5))^n+(1-sqrt(5))^n)/2; G.f.:(1-x)/(1-2x-4x^2); E.g.f.: 
               exp(x)cosh(sqrt(5)x).
%F A084057 a(2n+1)=2a(n)a(n+1)-(-4)^n. - Mario Catalani (mario.catalani(AT)unito.it), 
               Jun 13 2003
%F A084057 a(n)=sum{k=0..floor(n/2), binomial(n, 2k)5^k }. - Paul Barry (pbarry(AT)wit.ie), 
               Jul 25 2004
%F A084057 a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*5^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 26 2007
%o A084057 sage: from sage.combinat.sloane_functions import recur_gen2b sage: it 
               = recur_gen2b(1,1,2,4, lambda n: 0) sage: [it.next() for i in xrange(1,
               26)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 09 2008
%o A084057 (Other) sage: [lucas_number2(n,2,-4)/2 for n in xrange(0, 26)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A084057 Cf. A046717, A002533.
%Y A084057 Equals (1/2) A087131.
%Y A084057 The following sequences (and others) belong to the same family: A001333, 
               A000129, A026150, A002605, A046717, A015518, A084057, A063727, A002533, 
               A002532, A083098, A083099, A083100, A015519.
%Y A084057 A143969 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 06 2008]
%Y A084057 Sequence in context: A026086 A032282 A163302 this_sequence A091649 A125628 
               A078672
%Y A084057 Adjacent sequences: A084054 A084055 A084056 this_sequence A084058 A084059 
               A084060
%K A084057 easy,nonn
%O A084057 0,3
%A A084057 Paul Barry (pbarry(AT)wit.ie), May 10 2003