2,417 research outputs found
The golden ratio, Fibonacci numbers and BBP-type formulas
We derive interesting arctangent identities involving the golden ratio,
Fibonacci numbers and Lucas numbers. Binary BBP-type formulas for the
arctangents of certain odd powers of the golden ratio are also derived, for the
first time in the literature. Finally we derive golden-ratio-base BBP-type
formulas for some mathematical constants, including , ,
and . The nary BBP-type formulas derived here
are considerably simpler than similar results contained in earlier literature
Partial sums and generating functions for powers of second order sequences with indices in arithmetic progression
The sums , , and are evaluated; where is any positive integer, ,
and are any arbitrary integers, is arbitrary, and
are the Lucas sequences of the first kind, and of the second kind,
respectively; and is the Horadam sequence. Pantelimon St\uanic\ua set
out to evaluate the sum . His solution is not
complete because he made the assumption that , thereby giving
effectively only the partial sum for , the Lucas sequence of the first
kind.Comment: 8 pages, no figures, no table
A master identity for Horadam numbers
We derive an identity involving Horadam numbers. Numerous new identities as
well as those found in the existing literature are subsumed in this single
identity.Comment: 14 pages, no figures, no table
A novel approach to the discovery of binary BBP-type formulas for polylogarithm constants
Using a clear and straightforward approach, we discover and prove new binary
digit extraction BBP-type formulas for polylogarithm constants. Some known
results are also rediscovered in a more direct and elegant manner. Numerous
experimentally discovered and previously unproved binary BBP-type formulas are
also proved
Infinite arctangent sums involving Fibonacci and Lucas numbers
Using a straightforward elementary approach, we derive numerous infinite
arctangent summation formulas involving Fibonacci and Lucas numbers. While most
of the results obtained are new, a couple of celebrated results appear as
particular cases of the more general formulas derived here
Some remarkable infinite product identities involving Fibonacci and Lucas numbers
By applying the classic telescoping summation formula and its variants to
identities involving inverse hyperbolic tangent functions having inverse powers
of the golden ratio as arguments and employing subtle properties of the
Fibonacci and Lucas numbers, we derive interesting general infinite product
identities involving these numbers.Comment: 11 pages, corrected a typ
A non-PSLQ route to BBP-type formulas
BBP-type formulas are usually discovered experimentally, through computer
searches. In this paper, however, starting with two simple generators, and
hence without doing any computer searches, we derive a wide range of BBP-type
formulas in general bases. Many previously discovered BBP-type formulas turn
out to be particular cases of the formulas derived here
Summation Identities Involving Padovan and Perrin Numbers
Unlike in the case of Fibonacci and Lucas numbers, there is a paucity of
literature dealing with summation identities involving the Padovan and Perrin
numbers. In this paper, we derive various summation identities for these
numbers, including binomial and double binomial identities. Our results derive
from the rich algebraic properties exhibited by the zeros of the characteristic
polynomial of the Padovan/Perrin sequence.Comment: 18 pages, no figures, made a typographical correctio
A new Fibonacci identity and its associated summation identities
We derive a new Fibonacci identity. This single identity subsumes important
known identities such as those of Catalan, Ruggles, Halton and others, as well
as standard general identities found in the books by Vajda, Koshy and others.
We also derive several binomial and ordinary summation identities arising from
this identity; in particular we obtain a generalization of Halton's general
Fibonacci identity.Comment: 12 pages, no figure
A novel approach to the discovery of ternary BBP-type formulas for polylogarithm constants
Using a clear and straightforward approach, we prove new ternary (base 3)
digit extraction BBP-type formulas for polylogarithm constants. Some known
results are also rediscovered in a more direct and elegant manner. A previously
unproved degree~4 ternary formula is also proved. Finally, a couple of ternary
zero relations are established, which prove two known but hitherto unproved
formulas
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