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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)+... +0
5
1, 1, 3, 8, 21, 55, 144, 377, 987, 2584, 6765, 17711, 46368, 121393, 317811, 832040, 2178309, 5702887, 14930352, 39088169, 102334155, 267914296, 701408733, 1836311903, 4807526976, 12586269025, 32951280099, 86267571272, 225851433717 (list; graph; listen)
OFFSET

0,3

COMMENT

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

FORMULA

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

EXAMPLE

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

MAPLE

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

CROSSREFS

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

Sequence in context: A027932 A084625 A001906 this_sequence A072632 A001671 A090413

Adjacent sequences: A088302 A088303 A088304 this_sequence A088306 A088307 A088308

KEYWORD

easy,nonn

AUTHOR

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

EXTENSIONS

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

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

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Last modified July 25 07:41 EDT 2008. Contains 142293 sequences.


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