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Autor:
Dombrowski, Muhammad Adam
This paper proves why the Stirling numbers show up in a experimentally determined formula for the $k$-bonaccis. We develop a bijection between a previously determined summation formula for $k$-bonaccis and an experimentally determined formula, proven
Externí odkaz:
http://arxiv.org/abs/2407.18355
Publikováno v:
EPTCS 403, 2024, pp. 49-53
We consider Dyck paths having height at most two with some constraints on the number of consecutive valleys at height one which must be followed by a suitable number of valleys at height zero. We prove that they are enumerated by so-called Q-bonacci
Externí odkaz:
http://arxiv.org/abs/2406.16394
Autor:
Dovgal, Sergey, Kirgizov, Sergey
A binary word is called $q$-decreasing, for $q>0$, if every of its length maximal factors of the form $0^a1^b$, $a>0$, satisfies $q \cdot a > b$. We bijectively link $q$-decreasing words with certain prefixes of the cutting sequence of the line $y=qx
Externí odkaz:
http://arxiv.org/abs/2310.01213
Autor:
Pain, Jean-Christophe
In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a known expr
Externí odkaz:
http://arxiv.org/abs/2211.00364
Autor:
Parks, Harold R., Wills, Dean C.
We present an elementary proof of the generalization of the $k$-bonacci Binet formula, a closed form calculation of the $k$-bonacci numbers using the roots of the characteristic polynomial of the $k$-bonacci recursion.
Comment: 4 pages
Comment: 4 pages
Externí odkaz:
http://arxiv.org/abs/2208.06989
Autor:
Parks, Harold R., Wills, Dean C.
We give a combinatorial proof of a formula giving the partial sums of the $k$-bonacci sequence as alternating sums of powers of two multiplied by binomial coefficients. As a corollary we obtain a formula for the $k$-bonacci numbers.
Comment: 6 p
Comment: 6 p
Externí odkaz:
http://arxiv.org/abs/2208.01224
Akademický článek
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Autor:
Kirgizov, Sergey
We present a quite curious generalization of multi-step Fibonacci numbers. For any positive rational $q$, we enumerate binary words of length $n$ whose maximal factors of the form $0^a1^b$ satisfy $a = 0$ or $aq > b$. When $q$ is an integer we redisc
Externí odkaz:
http://arxiv.org/abs/2201.00782
A word $w$ is said to be closed if it has a proper factor $x$ which occurs exactly twice in $w$, as a prefix and as a suffix of $w$. Based on the concept of Ziv-Lempel factorization, we define the closed $z$-factorization of finite and infinite words
Externí odkaz:
http://arxiv.org/abs/2106.03202
Akademický článek
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