%I A063727
%S A063727 1,2,8,24,80,256,832,2688,8704,28160,91136,294912,954368,3088384,
%T A063727 9994240,32342016,104660992,338690048,1096024064,3546808320,
%U A063727 11477712896,37142659072,120196169728,388962975744,1258710630400
%N A063727 a(n) = 2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.
%C A063727 Convergents to golden ratio (1+sqrt(5))/2.
%C A063727 Number of ways to tile an n-board with two types of colored squares and 
               four types of colored dominoes.
%C A063727 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
%D A063727 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, id. 235.
%D A063727 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%H A063727 Harry J. Smith, <a href="b063727.txt">Table of n, a(n) for n=0,...,200</
               a>
%H A063727 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A063727 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A063727 a(n) = sqrt(5)/10*((1+sqrt(5))^(n+1)-(1-sqrt(5))^(n+1)). G.f.: 1/(1-2*x-4*x^2). 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 16 2001
%F A063727 a(2n)=4a(n-1)^2+a(n)^2. A084057(n+1)/a(n) converges to sqrt(5). - Mario 
               Catalani (mario.catalani(AT)unito.it), Jun 13 2003
%F A063727 E.g.f. : exp(x)(cosh(sqrt(5)x))+sinh(sqrt(5)x)/sqrt(5)) - Paul Barry 
               (pbarry(AT)wit.ie), Sep 20 2003
%F A063727 a(n) = 2^n*Fibonacci(n+1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 
               25 2003
%F A063727 a(n)=sum{k=0..floor(n/2), C(n, 2k+1)5^k} - Paul Barry (pbarry(AT)wit.ie), 
               Nov 15 2003
%F A063727 a(n)=U(n, i/sqrt(4))(-i*sqrt(4))^n, i^2=-1 - Paul Barry (pbarry(AT)wit.ie), 
               Nov 17 2003
%F A063727 Simplified formula: ((1+sqrt5)^n-(1-sqrt5)^n)/sqrt20. Offset 1. a(3)=8 
               [From Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009]
%F A063727 a(n)=first binomial trnasform of 1,1,5,5,25,25 [From Al Hakanson (hawkuu(AT)gmail.com), 
               Jul 20 2009]
%p A063727 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+4*a[n-2]od: seq(a[n], 
               n=1..33);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 
               15 2008]
%o A063727 (PARI) s(n)=if(n<2,n+1,(s(n-1)+(s(n-2)*2))*2); for(n=0,32,print(s(n)))
%o A063727 sage: from sage.combinat.sloane_functions import recur_gen2b sage: it 
               = recur_gen2b(1,2,2,4, lambda n: 0) sage: [it.next() for i in xrange(1,
               26)] - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jul 03 2008
%o A063727 (Other) sage: [lucas_number1(n,2,-4) for n in xrange(1, 26)] # [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 22 2009]
%o A063727 (PARI) { for (n=0, 200, if (n>1, a=2*a1 + 4*a2; a2=a1; a1=a, if (n, a=a1=2, 
               a=a2=1)); write("b063727.txt", n, " ", a) ) } [From Harry J. Smith 
               (hjsmithh(AT)sbcglobal.net), Aug 28 2009]
%Y A063727 Essentially the same as A085449.
%Y A063727 Equals 2 * A087206(n+1). Cf. A006483.
%Y A063727 Row sums of triangle A016095.
%Y A063727 Cf. A103435.
%Y A063727 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 A063727 Sequence in context: A006952 A034741 A085449 this_sequence A127362 A133443 
               A094038
%Y A063727 Adjacent sequences: A063724 A063725 A063726 this_sequence A063728 A063729 
               A063730
%K A063727 nonn
%O A063727 0,2
%A A063727 Klaus E. Kastberg (kastberg(AT)hotkey.net.au), Aug 12 2001
%E A063727 Better description from Jason Earls (zevi_35711(AT)yahoo.com) and Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Aug 16 2001