%I A046717
%S A046717 1,1,5,13,41,121,365,1093,3281,9841,29525,88573,265721,797161,2391485,
%T A046717 7174453,21523361,64570081,193710245,581130733,1743392201,5230176601,
%U A046717 15690529805,47071589413,141214768241,423644304721,1270932914165
%N A046717 a(n) = 2a(n-1)+3a(n-2), a(0) = a(1) = 1.
%C A046717 Form the digraph with matrix A=[0,1,1,1;1,0,1,1;1,1,0,1;1,0,1,1]. Then 
               the sequence 0,1,1,5,... or (3^(n-1)-(-1)^n)/2+0^n/3 with g.f. x(1-x)/
               (1-2x-3x^2) corresponds to the (1,2) term of A^n. - Paul Barry (pbarry(AT)wit.ie), 
               Oct 02 2004
%C A046717 3*a(n+1) + a(n) = 4*A060925(n); a(n+1) = A015518(n) + A060925(n); a(n+1) 
               - 6*A015518(n) = (-1)^n. The floretion -.5'i + 'k - .5i' - .5k' + 
               .5'ii' + 1.5'jj' - .5'kk' - .5'ij' + 'ki' - .5e is a generator of 
               the above formula. - Creighton Dement (creighton.k.dement(AT)uni-oldenburg.de), 
               Nov 15 2004
%C A046717 The sequence corresponds to the (1,1) term of the matrix [1,2;2,1]^n. 
               - Simone Severini (ss54(AT)york.ac.uk), Dec 04 2004
%C A046717 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 4 times the bottom to get the new top. The limit of the 
               sequence of fractions is 2. - Cino Hilliard (hillcino368(AT)gmail.com), 
               Sep 25 2005
%C A046717 a(n)^2 + (2*A015518(n))^2 = a(2n). E.g. a(3) = 13, 2*A015518(3) = 14, 
               A046717(6) = 365. 13^2 + 14^2 = 365. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Jun 17 2006
%D A046717 John Derbyshire, Prime Obsession, Joseph Henry Press, April 2004, see 
               p. 16.
%H A046717 C. Banderier and D. Merlini, <a href="http://www.dsi.unifi.it/~merlini/
               poster.ps">Lattice paths with an infinite set of jumps</a>, FPSAC02, 
               Melbourne, 2002.
%H A046717 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%F A046717 G.f.: (1-x)/((1+x)*(1-3*x)). a(n)=(3^n+(-1)^n)/2.
%F A046717 a(n)=Sum{k=0..n, Binomial(n, 2k)2^(2k)} - Paul Barry (pbarry(AT)wit.ie), 
               Feb 26 2003
%F A046717 Binomial transform of A000302 (powers of 4) with interpolated zeros. 
               Inverse binomial transform of A081294. - Paul Barry (pbarry(AT)wit.ie), 
               Mar 17 2003
%F A046717 E.g.f.: exp(x)cosh(2x). - Paul Barry (pbarry(AT)wit.ie), Mar 17 2003
%F A046717 a(n)=ceiling(3^n/4)+floor(3^n/4)=ceiling(3^n/4)^2-floor(3^n/4)^2. - Paul 
               Barry (pbarry(AT)wit.ie), Jan 17 2005
%F A046717 a(n)=sum{k=0..n, sum{j=0..n, C(n,j)C(n-j,k)*(1+(-1)^(j-k))/2}}; - Paul 
               Barry (pbarry(AT)wit.ie), May 21 2006
%F A046717 a(n)=Sum_{k, 0<=k<=n}A098158(n,k)*4^(n-k). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Dec 26 2007
%F A046717 a(n) = (3^n+(-1)^n)/2. - M. F. Hasler, Mar 20 2008
%F A046717 a(n) = 2 A015518(n) + (-1)^n ; for n>0, a(n) = A080925(n). - M. F. Hasler, 
               Mar 20 2008
%F A046717 ((1+sqrt4)^n+(1-sqrt4)^n)/2. The offset is 0. a(3)=13. [From Al Hakanson 
               (hawkuu(AT)gmail.com), Nov 22 2008]
%p A046717 a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=2*a[n-1]+3*a[n-2] od: seq(a[n], 
               n=0..33);# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 
               14 2008]
%t A046717 Expand[Table[((3)^n + (-1)^n)/(2), {n, 0, 30}]] - Artur Jasinski (grafix(AT)csl.pl), 
               Dec 10 2006
%o A046717 (PARI) if(n<0,0,(3^n+(-1)^n)/2)
%o A046717 (Other) sage: [lucas_number2(n,2,-3)/2 for n in xrange(0, 27)]# [From 
               Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 30 2009]
%Y A046717 The first difference sequence of A015518.
%Y A046717 Row sums of triangle A080928.
%Y A046717 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 A046717 Cf. A015518.
%Y A046717 Sequence in context: A100210 A080267 A034735 this_sequence A080925 A164907 
               A085601
%Y A046717 Adjacent sequences: A046714 A046715 A046716 this_sequence A046718 A046719 
               A046720
%K A046717 nonn,easy
%O A046717 0,3
%A A046717 G. Deroo (gervais.deroo(AT)lemel.fr) and M. Deroo.
%E A046717 Description corrected by and more terms from Michael Somos