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A099919
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F(3) + F(6) + F(9) +...+ F(3n), F(n) = Fibonacci numbers A000045.
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9
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2, 10, 44, 188, 798, 3382, 14328, 60696, 257114, 1089154, 4613732, 19544084, 82790070, 350704366, 1485607536, 6293134512, 26658145586, 112925716858, 478361013020, 2026369768940, 8583840088782, 36361730124070
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Could be defined as the partial sum of the even Fibonacci numbers, including a(0)=0. (*From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com),28 Nov 2010*)
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REFERENCES
| A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 25.
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FORMULA
| a(n) = (Fibonacci(3*n+2) - 1)/2 = (A015448(n+1)-1)/2. G.f.: 2/((1-x)*(1-4*x-x^2)).
a(n) = 2*A049652(n).
a(n)=sum_{0<=j<=i<=n} binomial(i, j)*F(i+j) - Benoit Cloitre benoit7848c(AT)orange.fr), May 21 2005
a(n)= 4*a(n-1)+a(n-2)+2, n>1 [From Gary Detlefs (gdetlefs(At)aol.com), Dec 8 2010]
a(n)=5*a(n-1)-3*a(n-2)-a(n-3), n>2, [From Gary Detlefs (gdetlefs(At)aol.com), Dec 8 2010]
a(n)= (fibonacci(3*n+3)+fibonacci(3*n)-2)/4. [From Gary Detlefs (gdetlefs(At)aol.com), Dec 8 2010]
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MATHEMATICA
| s=0; lst={s}; Do[f=Fibonacci[n]; If[EvenQ[f], AppendTo[lst, s+=f]], {n, 0, 5!}]; lst (*From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), 28 Nov 2010*)
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CROSSREFS
| Partial sums of A014445. Cf. A004794.
Cf. A087635.
Case k=3 of partial sums of fibonacci(k*n): A000071, A027941, A058038, A138134, A053606
Sequence in context: A122932 A080069 A068551 * A100397 A084059 A084609
Adjacent sequences: A099916 A099917 A099918 * A099920 A099921 A099922
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KEYWORD
| nonn
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AUTHOR
| Ralf Stephan, Oct 30 2004
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