0,3

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

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

Row sums of triangle A128908 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007

Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008: (Start)

Let P = the partial sum operator, A000012: (1; 1,1; 1,1,1;...) and A153463

= M, the partial sum & shift operator. It appears that beginning with any

randomly taken sequence S(n), iterates of the operations M * S(n), -> M * ANS,

-> P * ANS,...etc, (or starting with P) will rapidly converge upon a two-

sequence limit cycle of (1, 2, 5, 13, 34,...) and (1, 1, 3, 8, 21,...). (End)

a(0)=1, a(n) = (h^(2n) - h^(-2n))/sqrt(5), where h = (1+sqrt(5))/2.

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

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

a(n)= [((3+sqrt5)/2)^n-((3-sqrt5)/2)^n]/sqrt5 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Sep 23 2008]

a(5)=55=1*21+2*8+3*3+4*1+5*1=21+16+9+4+5

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

Cf. A000045. Apart from initial term, same as A001906.

A153463 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 27 2008]

Adjacent sequences: A088302 A088303 A088304 this_sequence A088306 A088307 A088308

Sequence in context: A027932 A084625 A001906 this_sequence A072632 A001671 A090413

easy,nonn

Miklos Kristof (kristmikl(AT)freemail.hu), Nov 05 2003

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Nov 06 2003

Further terms from Edward Scheinerman (ers(AT)jhu.edu), Feb 28 2007