Fibonacci and Lucas Numbers

@inproceedings{Hilton1997FibonacciAL,
  title={Fibonacci and Lucas Numbers},
  author={Peter J. Hilton and Derek A. Holton and Jean J. Pedersen},
  year={1997}
}
Consider the following number trick–try it out on your friends. You ask them to write down the numbers from 0 to 9. Against 0 and 1 they write any two numbers (we suggest two fairly small positive integers just to avoid tedious arithmetic, but all participants should write the same pair of numbers). Then against 2 they write the sum of the entries against 0 and 1; against 3 they write the sum of the entries against 1 and 2; and so on. Once they have completed the process, producing entries… 

ONE PARAMETER GENERALIZATIONS OF THE FIBONACCI AND LUCAS NUMBERS

We give one parameter generalizations of the Fibonacci and Lucas numbers denoted by {Fn(�)} and {Ln(�)}, respectively. We evaluate the Hankel determinants with entries {1/Fj+k+1(�) : 0 � i,jn} and

Combinatorial proofs of Honsberger-type identities

In this article, we consider some generalizations of Fibonacci numbers. We consider k-Fibonacci numbers (that follow the recurrence rule F k,n + 2 = kF k,n + 1 + F k,n ), the (k,ℓ)-Fibonacci numbers

ON THE INTEGRITY OF CERTAIN FIBONACCI SUMS

[n is an arbitrary natural number, r is an arbitrary (nonzero) real quantity) gives & positive integer k. Since both r and k turn out to be Fibonacci number ratios, the results established in this

THE FIRST 330 TERMS OF SEQUENCE A013583

Combinatorially, each term R(N)x counts the R(N) partitions of N into distinct Fibonacci numbers. Some of the recursion properties of this sequence are investigated in [2]. The difficulties in

On the complex k-Fibonacci numbers

We first study the relationship between the k-Fibonacci numbers and the elements of a subset of . Later, and since generally studies that are made on the Fibonacci sequences consider that these

Random Approaches to Fibonacci Identities

By adopting a probabilistic viewpoint, many of the remaining identities in the Fibonacci identities can be explained combinatorially, and even the “irrational-looking” Binet’s formula for the n-th fibonacci number is demonstrated.

Extremal orders of the Zeckendorf sum of digits of powers

Denote by s_F(n) the minimal number of Fibonacci numbers needed to write n as a sum of Fibonacci numbers. We obtain the extremal minimal and maximal orders of magnitude of s_F(n^h)/s_F(n) for any h>=

GCD of sums of $k$ consecutive Fibonacci, Lucas, and generalized Fibonacci numbers

We explore the sums of k consecutive terms in the generalized Fibonacci sequence ( G n ) n ≥ 0 given by the recurrence G n = G n − 1 + G n − 2 for all n ≥ 2 with integral initial conditions G 0 and G

Sums of certain products of Fibonacci and Lucas numbers

In Section 2 we prove a theorem involving a sum of products of Fibonacci numbers, and in Section 3 we prove the corresponding theorem for the Lucas numbers. In Section 4 we present three additional

INITIAL DIGITS IN NUMBER THEORY

  • Mathematics
  • 2010
1. J. C. Butcher. "On a Conjecture Concerning a Set of Sequences Satisfying The Fibonacci Difference Equation/ The Fibonacci Quarterly 16 (1978):8183. 2. M. D. Hendy, "Stolarskys Distribution of
...

References

SHOWING 1-3 OF 3 REFERENCES

Combinatorial Expressions for Lucas Numbers.

  • The Fibonacci Quarterly
  • 1998

Chebyshev polynomials of the second kind

On #1*-Order Linear Recurrences.

  • The Fibonacci Quarterly
  • 1985