login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A056014 (Fibonacci(2n-1)-Fibonacci(n+1))/2 . 6
0, 0, 0, 1, 4, 13, 38, 106, 288, 771, 2046, 5401, 14212, 37324, 97904, 256621, 672336, 1760997, 4611642, 12075526, 31617520, 82781215, 216732890, 567428401, 1485570024, 3889310328, 10182407328, 26657986681, 69791674108 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

With a(0)=0, a(1)=1, a(2)=1, a(3)=2, this recurrence produces a(n)=A000045(n) (Fibonacci numbers).

Number of (s(0), s(1), ..., s(n)) such that 0 < s(i) < 5 and |s(i) - s(i-1)| <= 1 for i = 1,2,....,n, s(0) = 1, s(n) = 4. - Herbert Kociemba, Jun 16 2004

REFERENCES

É. Czabarka, R. Flórez, L. Junes, A Discrete Convolution on the Generalized Hosoya Triangle, Journal of Integer Sequences, 18 (2015), #15.1.6.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-3,-2,1)

FORMULA

a(n)=4a(n-1)-3a(n-2)-2a(n-3)+a(n-4), a(0)=a(1)=a(2)=0, a(3)=1.

Convolution of Fibonacci numbers F(n) with F(2n). - Benoit Cloitre, Jun 07 2004

G.f.: x^3/((1-x-x^2)*(1-3*x+x^2)) - Herbert Kociemba, Jun 16 2004

Binomial transform of x^3/(1-3x^2+x^4), or (essentially) F(2n) with interpolated zeros. a(n)=sum{k=0..n, binomial(n, k)((3/2-sqrt(5)/2)^(k/2)((sqrt(5)/20+1/4)(-1)^k-sqrt(5)/20-1/4)+ (sqrt(5)/2+3/2)^(k/2)((sqrt(5)/20-1/4)(-1)^k-sqrt(5)/20+1/4))} - Paul Barry, Jul 26 2004

Convolution of the powers of 2 (A000079) with the number of positive rational knots with 2n+1 crossings (A051450), with three leading zeros. - Graeme McRae, Jun 28 2006

a(n) = (A001519(n)-A000045(n+1))/2. - R. J. Mathar, Jun 24 2011

MATHEMATICA

Table[(Fibonacci[2n-1]-Fibonacci[n+1])/2, {n, 0, 40}]  (* Harvey P. Dale, Mar 24 2011 *)

LinearRecurrence[{4, -3, -2, 1}, {0, 0, 0, 1}, 40] (* Vincenzo Librandi, Jun 23 2012

PROG

(PARI) a(n)=(fibonacci(2*n-1)-fibonacci(n+1))/2

(MAGMA) I:=[0, 0, 0, 1]; [n le 4 select I[n] else 4*Self(n-1)-3*Self(n-2)-2*Self(n-3)+Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 23 2012

CROSSREFS

Cf. A000045, A056015.

a(1-2n)=A059512(2n), a(-2n)=A027994(2n-1).

Sequence in context: A049611 A084851 A094706 * A247287 A159036 A058693

Adjacent sequences:  A056011 A056012 A056013 * A056015 A056016 A056017

KEYWORD

nonn,easy

AUTHOR

Asher Auel (asher.auel(AT)reed.edu), Jun 06 2000

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 05:25 EST 2017. Contains 294853 sequences.