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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

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Last modified May 24 22:11 EDT 2017. Contains 287008 sequences.