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A015448
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a(0)=1, a(1)=1, and a(n) = 4*a(n-1) + a(n-2) for n>=2.
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37
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1, 1, 5, 21, 89, 377, 1597, 6765, 28657, 121393, 514229, 2178309, 9227465, 39088169, 165580141, 701408733, 2971215073, 12586269025, 53316291173, 225851433717, 956722026041, 4052739537881, 17167680177565, 72723460248141, 308061521170129, 1304969544928657, 5527939700884757
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OFFSET
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0,3
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COMMENTS
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a(n) = A167808(3*n-1) for n>0. [From Reinhard Zumkeller, Nov 12 2009]
If one deletes the leading 0 in A084326, takes the inverse binomial transform, and adds a(0)=1 in front, one obtains this sequence here. [From Al Hakanson (hawkuu(AT)gmail.com), May 02 2009]
For n>=1, row sums of triangle
m/k.|..0.....1.....2.....3.....4.....5.....6.....7
==================================================
.0..|..1
.1..|..1.....4
.2..|..1.....4....16
.3..|..1.....8....16......64
.4..|..1.....8....48......64...256
.5..|..1....12....48.....256...256...1024
.6..|..1....12....96.....256..1280...1024...4096
.7..|..1....16....96.....640..1280...6144...4096...16384
which is triangle for numbers 4^k*C(m,k) with duplicated diagonals. - Vladimir Shevelev, Apr 12 2012
a(n) = a(n;-2) = 3^n sum_{k=0..n} C(n,k) F(k+1) (-2/3)^k, where a(n;d), n=0,1,..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al papers). We note that (see A033887) F(3n+1)=3^n a(n,2/3) = sum_{k=0..n} C(n,k) F(k-1) (-2/3)^k, which implies F(3n+1) + 3^(-n) a(n) = sum_{k=0}^{n} C(n,k) L(k) (-2/3)^k, where L(k) denote the k-th Lucas number. - Roman Witula, Jul 12 2012
a(n+1) is (for n>=0) the number of length-n strings of 5 letters {0,1,2,3,4} with no two adjacent nonzero letters identical. The general case (strings of L letters) is the sequence with g.f. (1+x)/(1-(L-1)*x-x^2). [Joerg Arndt, Oct 11 2012]
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REFERENCES
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D. Chmiela, K. Kaczmarek and R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted 2012).
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..200
Joerg Arndt, Fxtbook, pp.313-315
Tanya Khovanova, Recursive Sequences
R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042
Index entries for sequences related to linear recurrences with constant coefficients, signature (4,1).
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FORMULA
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a(n) = Fibonacci(3n-1) = ( (1+sqrt(5))*(2-sqrt(5))^n - (1-sqrt(5))*(2+sqrt(5))^n )/ (2*sqrt(5)).
O.g.f.: (1-3*x)/(1-4*x-x^2). - Len Smiley (smiley(AT)math.uaa.alaska.edu), Dec 09 2001
a(n) = Sum_{k, 0<=k<=n}3^k*A055830(n,k). - Philippe DELEHAM, Oct 18 2006
a(n) = upper left term in the 2 X 2 matrix [1,2; 2,3]^n - Gary W. Adamson, Mar 02 2008
[a(n), A001076(n)] = [1,4; 1,3]^n * [1,0] - Gary W. Adamson, Mar 21 2008
a(n) = Fibonacci(3n+1) mod Fibonacci(3n), n>0
a(n) = (A000032(3*n)-Fibonacci(3*n))/2 = (A014448(n)-A014445(n)/2.
For n>=2, a(n)=F_n(4)+F_(n+1)(4), where F_n(x) is Fibonacci polynomial (cf.A049310): F_n(x)=sum{i=0,...,floor((n-1)/2)}C(n-i-1,i)x^(n-2*i-1). - Vladimir Shevelev, Apr 13 2012
a(n) = A001076(n+1) - 3*A001076(n). - R. J. Mathar, Jul 12 2012
From Gary Detlefs and Wolfdieter Lang, Aug 20 2012 (Start)
a(n) = (5*F(n)^3 + 5*F(n-1)^3 + 3*(-1)^n*F(n-2))/2,
a(n) = (F(n+1)^3 + 2*F(n)^3 - F(n-2)^3)/2, n >= 0, with F(-1) = 1 and F(-2) = -1. Second line from first one with 3*(-1)^n* F(n-2) = F(n-1)^3 - 4*F(n-2)^3 - F(n-3)^3 (in Koshy's book, p. 89, 32. (with a - sign) and 33. For the Koshy reference see A000045) and the F^3 recurrence (see row n=4 of A055870, or Koshy p. 87, 1.). First line from the preceding R. J. Mathar formula with F(3*n) = 5*F(n)^3 + 3*(-1)^n*F(n) (Koshy p.89, 46.) and the above mentioned formula, Koshy's 32. and 33., with n -> n+2 in order to eliminate F(n+1)^3. (End)
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MAPLE
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with(combinat): a:=n->fibonacci(n, 4)-3*fibonacci(n-1, 4): seq(a(n), n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2008
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MATHEMATICA
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Fibonacci/@(3*Range[30]-1) [From Vladimir Joseph Stephan Orlovsky, Mar 01 2010]
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PROG
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(Maxima)
a[0]:1$
a[1]:1$
a[n]:=4*a[n-1]+a[n-2]$
A015448(n):=a[n]$
makelist(A015448(n), n, 0, 30); /* Martin Ettl, Nov 03 2012 */
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CROSSREFS
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Cf. A001076, A147722 (INVERT transform), A109499 (INVERTi transform), A154626 (Binomial transform), A086344 (inverse binomial transform)
Sequence in context: A019992 A010917 A099843 * A035011 A113987 A188707
Adjacent sequences: A015445 A015446 A015447 * A015449 A015450 A015451
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KEYWORD
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nonn,easy
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AUTHOR
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Olivier Gérard
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STATUS
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approved
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