Search: id:A088305
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%I A088305
%S A088305 1,1,3,8,21,55,144,377,987,2584,6765,17711,46368,121393,317811,832040,
%T A088305 2178309,5702887,14930352,39088169,102334155,267914296,701408733,
%U A088305 1836311903,4807526976,12586269025,32951280099,86267571272,225851433717
%N A088305 a(0)=1, a(n)=F(2n) where F(n) = Fibonacci numbers A000045. Has the property: 
               a(n)=1*a(n-1)+2*a(n-2)+3*a(n-3)+4*a(n-4)+...
%C A088305 F(2n)=1*F(2n-2)+2*F(2n-4)+3*F(2n-6)+4*F(2n-8)+... F(2n+1)=1+1*F(2n-1)+2*F(2n-3)+3*F(2n-5)+4*F(2n-7)+... 
               Convolution with 1,3,6,10,...n(n+1)/2: 1*F(2n)+3*F(2n-2)+6*F(2n-4)+10*F(2n-6)+...=F(2n+3)-1 
               1*F(2n-1)+3*F(2n-3)+6*F(2n-5)+10*F(2n-7)+...=F(2n+2)-n-1
%C A088305 Also the number of spanning trees of a graph formed by joining a single 
               vertex to all vertices of a path on n-1 vertices. - Edward Scheinerman 
               (ers(AT)jhu.edu), Feb 28 2007
%C A088305 Row sums of triangle A128908 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 21 2007
%C A088305 Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008: 
               (Start)
%C A088305 Let P = the partial sum operator, A000012: (1; 1,1; 1,1,1;...) and A153463
%C A088305 = M, the partial sum & shift operator. It appears that beginning with 
               any
%C A088305 randomly taken sequence S(n), iterates of the operations M * S(n), -> 
               M * ANS,
%C A088305 -> P * ANS,...etc, (or starting with P) will rapidly converge upon a 
               two-
%C A088305 sequence limit cycle of (1, 2, 5, 13, 34,...) and (1, 1, 3, 8, 21,...). 
               (End)
%F A088305 a(0)=1, a(n) = (h^(2n) - h^(-2n))/sqrt(5), where h = (1+sqrt(5))/2.
%F A088305 a(n)=Sum{k=1..n+1} binomial(n+k-1,n-k), with a(0)=1. - Paolo P. Lava 
               (ppl(AT)spl.at), Apr 13 2007
%F A088305 a(0)=1, a(1)=1, a(2)=3, a(n+1)=3*a(n)-a(n-1) for n>=2 . G.f.: (1-2x+x^2)/
               (1-3x+x^2). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007
%F A088305 a(n)= [((3+sqrt5)/2)^n-((3-sqrt5)/2)^n]/sqrt5 [From Geoffrey Critzer 
               (critzer.geoffrey(AT)usd443.org), Sep 23 2008]
%e A088305 a(5)=55=1*21+2*8+3*3+4*1+5*1=21+16+9+4+5
%p A088305 with(combstruct): ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), b): ZL1:=Prod(begin_blockP, 
               Z, end_blockP): ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, 
               Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, 
               Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL1, ZL2), b=ZL1], 
               ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, 
               end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, 
               mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, 
               {Q}, unlabelled], size=n), n=1..29); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 08 2008
%Y A088305 Cf. A000045. Apart from initial term, same as A001906.
%Y A088305 A153463 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]
%Y A088305 Sequence in context: A027932 A084625 A001906 this_sequence A072632 A001671 
               A090413
%Y A088305 Adjacent sequences: A088302 A088303 A088304 this_sequence A088306 A088307 
               A088308
%K A088305 easy,nonn
%O A088305 0,3
%A A088305 Miklos Kristof (kristmikl(AT)freemail.hu), Nov 05 2003
%E A088305 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 06 
               2003
%E A088305 Further terms from Edward Scheinerman (ers(AT)jhu.edu), Feb 28 2007

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