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A033887
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Fibonacci(3n+1).
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26
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1, 3, 13, 55, 233, 987, 4181, 17711, 75025, 317811, 1346269, 5702887, 24157817, 102334155, 433494437, 1836311903, 7778742049, 32951280099, 139583862445, 591286729879, 2504730781961, 10610209857723
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Second binomial transform of (1,1,5,5,25,25,....). - Paul Barry (pbarry(AT)wit.ie), Jul 16 2003
a(n) = A167808(3*n+1). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Nov 12 2009]
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 22 2010: (Start)
Equals INVERT transform of A104934: (1, 2, 8, 28, 100, 356,...) and INVERTi
transform of A005054: (1, 4, 20, 100, 500,...). (End)
a(n) is the number of compositions of n when there are 3 types of 1 and 4 types of other natural numbers. [From Milan R. Janjic (agnus(AT)blic.net), Aug 13 2010]
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LINKS
| Index entries for sequences related to linear recurrences with constant coefficients
Tanya Khovanova, Recursive Sequences
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FORMULA
| a(n) = A001076(n)+A001077(n).
a(n) = 2*A049651(n) + 1.
a(n) = 4a(n-1)+a(n-2), n>1, a(0)=1, a(1)=3; G.f.: (1-x)/(1-4*x-x^2); a(n)=[ (1+sqrt(5))(2+sqrt(5))^n - (1-sqrt(5))(2-sqrt(5))^n ]/2*sqrt(5).
a(n)=sum{k=0..n, sum{j=0..n-k, C(n,j)C(n-j,k)F(n-j+1)}}; - Paul Barry (pbarry(AT)wit.ie), May 19 2006
First differences of A001076. [From Al Hakanson (hawkuu(AT)gmail.com), May 02 2009]
a(n)=sum{k=0..n, C(n,k)*F(n+k+1)}. [From Paul Barry (pbarry(AT)wit.ie), Apr 19 2010]
If p[1]=3, p[i]=4, (i>1), and if A is Hessenberg matrix of order n defined by: A[i,j]=p[j-i+1], (i<=j), A[i,j]=-1, (i=j+1), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det A. [From Milan R. Janjic (agnus(AT)blic.net), Apr 29 2010]
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MAPLE
| with(combinat): a:=n->fibonacci(n, 4)-fibonacci(n-1, 4): seq(a(n), n=1..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008
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MATHEMATICA
| Fibonacci[Range[1, 5!, 3]] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), May 18 2010]
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PROG
| (MAGMA) [Fibonacci(3*n +1): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011
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CROSSREFS
| Cf. A104934, A005054 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jul 22 2010]
Sequence in context: A140320 A037583 A093834 * A183804 A117376 A151318
Adjacent sequences: A033884 A033885 A033886 * A033888 A033889 A033890
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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