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A093040
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Expansion of (1+x)/((1+x+x^2)(1-x-x^2)).
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3
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1, 1, 1, 3, 4, 6, 11, 17, 27, 45, 72, 116, 189, 305, 493, 799, 1292, 2090, 3383, 5473, 8855, 14329, 23184, 37512, 60697, 98209, 158905, 257115, 416020, 673134, 1089155, 1762289, 2851443, 4613733, 7465176, 12078908, 19544085, 31622993, 51167077
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| The sequence 0,1,1,1,3... has a(n)=Fib(n+1)/2-A049347(n)/2. It counts paths of length n between two of the vertices of the graph with adjacency matrix [0,1,0,0;0,0,1,1;1,1,0,0;0,0,1,0].
Diagonal sums of Riordan array ((1+x), x(1+x)^2). - Paul Barry (pbarry(AT)wit.ie), May 31 2006
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REFERENCES
| MacKay, Information Theory, Inference and Learning Algorithms, CUP, 2003, p. 251
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LINKS
| A. Stoimenow, Generating Functions, Fibonacci Numbers and Rational Knots
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FORMULA
| G.f.: ((1+x)/(1-x-x^2)+(1-x^2)/(1-x^3))/2; a(n)=a(n-2)+2a(n-3)+a(n-5); a(n)=Fib(n+2)/2+sqrt(3)sin(2*pi*n/3+pi/3)/3=Fib(n+2)/2+A057078(n)/2.
a(n-1)=sum{k=0..floor(n/2), if(mod(n-k, 2)=1, binomial(n-k, k), 0)}; a(n-1)=A094686(n)-Fib(n); - Paul Barry (pbarry(AT)wit.ie), Jan 13 2005
a(n)=sum{k=0..floor(n/2), C(2k+1,n-2k)}; - Paul Barry (pbarry(AT)wit.ie), May 31 2006
a(n)=floor(Fibonacci(n+3)/2)-floor(Fibonacci(n+1)/2). [From Gary Detlefs (gdetlefs(AT)aol.com), Mar 13 2011]
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MATHEMATICA
| a=0; b=0; lst={a, b}; Do[z=a+b+1; AppendTo[lst, z]; a=b; b=z; z=a+b; AppendTo[lst, z]; a=b; b=z; z=a+b-1; AppendTo[lst, z]; a=b; b=z, {n, 40}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Feb 16 2010]
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CROSSREFS
| Cf. A005252.
Sequence in context: A154331 A001130 A069825 * A022935 A192813 A048229
Adjacent sequences: A093037 A093038 A093039 * A093041 A093042 A093043
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KEYWORD
| easy,nonn
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 15 2004
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