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A033887 a(n) = Fibonacci(3n+1). 37
1, 3, 13, 55, 233, 987, 4181, 17711, 75025, 317811, 1346269, 5702887, 24157817, 102334155, 433494437, 1836311903, 7778742049, 32951280099, 139583862445, 591286729879, 2504730781961, 10610209857723, 44945570212853, 190392490709135, 806515533049393, 3416454622906707 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A063727, and second binomial transform of (1,1,5,5,25,25,...), which is A074872 with offset 0. - Paul Barry, Jul 16 2003

a(n) = A167808(3*n+1). - Reinhard Zumkeller, Nov 12 2009

Equals INVERT transform of A104934: (1, 2, 8, 28, 100, 356, ...) and INVERTi transform of A005054: (1, 4, 20, 100, 500, ...). - Gary W. Adamson, Jul 22 2010

a(n) is the number of compositions of n when there are 3 types of 1 and 4 types of other natural numbers. - Milan Janjic, Aug 13 2010

F(3n+1) = 3^n*a(n;2/3), where a(n;d), n=0, 1, ..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also the papers by Witula et al.). - Roman Witula, Jul 12 2012

We note that the remark above by Paul Barry can be easily obtained from the following scaling identity for delta-Fibonacci numbers y^n a(n;x/y) = Sum_{k=0..n} binomial(n,k) (y-1)^(n-k) a(k;x) and the fact that a(n;2)=5^floor(n/2). Indeed, for x=y=2 we get 2^n a(n;1) = Sum_{k=0..n} binomial(n,k) a(k;2) and, by A000045: Sum_{k=0..n} binomial(n,k) 2^k a(k;1) = Sum_{k=0..n} binomial(n,k) F(k+1) 2^k = 3^n a(n;2/3) = F(3n+1). - Roman Witula, Jul 12 2012

Except for the first term, this sequence can be generated by Corollary 1 (iv) of Azarian's paper in the references for this sequence. - Mohammad K. Azarian, Jul 02 2015

Number of 1’s in the substitution system {0 -> 110, 1 -> 11100} at step n from initial string "1" (1 -> 11100 -> 111001110011100110110 -> ...). - Ilya Gutkovskiy, Apr 10 2017

REFERENCES

D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012).

LINKS

Indranil Ghosh, Table of n, a(n) for n = 0..1592

Mohammad K. Azarian, Fibonacci Identities as Binomial Sums, International Journal of Contemporary Mathematical Sciences, Vol. 7, No. 38, 2012, pp. 1871-1876.

P. Barry, A. Hennessy, The Euler-Seidel Matrix, Hankel Matrices and Moment Sequences, J. Int. Seq. 13 (2010) # 10.8.2, Example 13.

Tanya Khovanova, Recursive Sequences

R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042

Index entries for linear recurrences with constant coefficients, signature (4,1).

FORMULA

a(n) = A001076(n) + A001077(n) = A001076(n+1) - A001076(n).

a(n) = 2*A049651(n) + 1.

a(n) = 4*a(n-1) + a(n-2) for 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, May 19 2006

First differences of A001076. - Al Hakanson (hawkuu(AT)gmail.com), May 02 2009

a(n) = Sum_{k=0..n} C(n,k)*F(n+k+1). - Paul Barry, 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. - Milan Janjic, Apr 29 2010

a(n) = Sum_{i=0..n} C(n,n-i)*A063727(i). - Seiichi Kirikami, Mar 06 2012

a(n) = Sum_{k=0..n} A122070(n,k) = Sum_{k=0..n} A185384(n,k). - Philippe Deléham, Mar 13 2012

a(n) = A000045(A016777(n)). - Michel Marcus, Dec 10 2015

a(n) = F(2*n)*L(n+1) + F(n-1)*(-1)^n for n > 0. - J. M. Bergot, Feb 09 2016

a(n) = Sum_{k=0..n} binomial(n,k)*5^floor(k/2)*2^(n-k). - Tony Foster III, Sep 03 2017

2*a(n) = Fibonacci(3*n) + Lucas(3*n). - Bruno Berselli, Oct 13 2017

EXAMPLE

a(5) = Fibonacci(3*5 + 1) = Fibonacci(16) = 987. - Indranil Ghosh, Feb 04 2017

MAPLE

with(combinat): a:=n->fibonacci(n, 4)-fibonacci(n-1, 4): seq(a(n), n=1..22); # Zerinvary Lajos, Apr 04 2008

MATHEMATICA

Fibonacci[Range[1, 5!, 3]] (* Vladimir Joseph Stephan Orlovsky, May 18 2010 *)

PROG

(MAGMA) [Fibonacci(3*n+1): n in [0..100]]; // Vincenzo Librandi, Apr 17 2011

(PARI) a(n)=fibonacci(3*n+1) \\ Charles R Greathouse IV, Feb 03 2014

(PARI) Vec((1-x)/(1-4*x-x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

CROSSREFS

Cf. A000032, A000045, A104934, A005054, A063727 (inverse binomial transform), A082761 (binomial transform).

Sequence in context: A037583 A093834 A286191 * A291653 A183804 A117376

Adjacent sequences:  A033884 A033885 A033886 * A033888 A033889 A033890

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified November 20 04:05 EST 2017. Contains 294959 sequences.