Generalized Natural Density DF(Fk) of Fibonacci Word Article Swipe
Duaa Abdullah
,
Jasem Hamoud
·
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.26117/2079-6641-2025-52-3-7-23
YOU?
·
· 2025
· Open Access
·
· DOI: https://doi.org/10.26117/2079-6641-2025-52-3-7-23
This paper explores profound generalizations of the Fibonacci sequence, delving into random Fibonacci sequences, k-Fibonacci words, and their combinatorial properties. We established that the nth root of the absolute value of terms in a random Fibonacci sequence converges to 1.13198824 . . ., with subsequent refinements by Rittaud yielding a limit of approximately 1.20556943 for the expected value’s n-th root. Novel definitions, such as the natural density of sets of positive integers and the limiting density of Fibonacci sequences modulo powers of primes, provide a robust framework for our analysis. We introduce the concept of k-Fibonacci words, extending classical Fibonacci words to higher dimensions, and investigate their patterns alongside sequences like the Thue-Morse and Sturmian words. Our main results include a unique representation theorem for real numbers using Fibonacci numbers, a symmetry identity for sums involving Fibonacci words, $\sum_{k=1}^{b} \dfrac{(-1)^k F_a}{F_k F_{k+a}}= \sum_{k=1}^{a} \dfrac{(-1)^k F_b}{F_k F_{k+b}}$, and an infinite series identity linking Fibonacci terms to the golden ratio. These findings underscore the intricate interplay between number theory and combinatorics, illuminating the rich structure of Fibonacci-related sequences. В данной статье рассматриваются глубокие обобщения последовательности Фибоначчи, включая случайные последовательности Фибоначчи, k-слова Фибоначчи и их комбинаторные свойства. Мы установили, что корень n-й степени из абсолютного значения членов случайной последовательности Фибоначчи сходится к 1.13198824 . . ., с последующими уточнениями Ритто, дающими предел приблизительно 1.20556943 для корня n-й степени ожидаемого значения. Новые определения, такие как естественная плотность множеств положительных целых чисел и предельная плотность последовательностей Фибоначчи по модулю степеней простых чисел, обеспечивают надежную основу для нашего анализа. Мы вводим концепцию k-слов Фибоначчи, расширяя классические слова Фибоначчи до более высоких измерений, и исследуем их закономерности наряду с последовательностями, такими как слова Туэ-Морса и Штурма. Наши основные результаты включают теорему об уникальном представлении действительных чисел с помощью чисел Фибоначчи, тождество симметрии для сумм, содержащих слова Фибоначчи, $\sum_{k=1}^{b} \dfrac{(-1)^k F_a}{F_k F_{k+a}}= \sum_{k=1}^{a} \dfrac{(-1)^k F_b}{F_k F_{k+b}}$, и тождество бесконечного ряда, связывающее члены Фибоначчи с золотым сечением. Эти результаты подчёркивают сложную взаимосвязь теории чисел и комбинаторики, проливая свет на богатую структуру последовательностей, связанных с числами Фибоначчи.
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- Landing Page
- https://doi.org/10.26117/2079-6641-2025-52-3-7-23
- https://krasec.ru/wp-content/uploads/2025/11/Abdulah-2.pdf
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- OpenAlex ID
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https://openalex.org/W4416764183Canonical identifier for this work in OpenAlex
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https://doi.org/10.26117/2079-6641-2025-52-3-7-23Digital Object Identifier
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Generalized Natural Density DF(Fk) of Fibonacci WordWork title
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articleOpenAlex work type
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ruPrimary language
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2025Year of publication
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2025-11-27Full publication date if available
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Duaa Abdullah, Jasem HamoudList of authors in order
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| abstract_inverted_index.значения | 202 |
| abstract_inverted_index.множеств | 233 |
| abstract_inverted_index.надежную | 248 |
| abstract_inverted_index.основные | 280 |
| abstract_inverted_index.проливая | 327 |
| abstract_inverted_index.расширяя | 258 |
| abstract_inverted_index.степеней | 244 |
| abstract_inverted_index.сходится | 207 |
| abstract_inverted_index.Fibonacci-related | 174 |
| abstract_inverted_index.Туэ-Морса | 276 |
| abstract_inverted_index.значения. | 226 |
| abstract_inverted_index.свойства. | 193 |
| abstract_inverted_index.сечением. | 317 |
| abstract_inverted_index.Фибоначчи | 189, 206, 241, 261, 314 |
| abstract_inverted_index.исследуем | 267 |
| abstract_inverted_index.концепцию | 255 |
| abstract_inverted_index.обобщения | 181 |
| abstract_inverted_index.плотность | 232, 239 |
| abstract_inverted_index.связанных | 333 |
| abstract_inverted_index.симметрии | 294 |
| abstract_inverted_index.случайной | 204 |
| abstract_inverted_index.случайные | 185 |
| abstract_inverted_index.структуру | 331 |
| abstract_inverted_index.тождество | 293, 309 |
| abstract_inverted_index.Фибоначчи, | 183, 187, 257, 292, 299 |
| abstract_inverted_index.Фибоначчи. | 336 |
| abstract_inverted_index.измерений, | 265 |
| abstract_inverted_index.ожидаемого | 225 |
| abstract_inverted_index.предельная | 238 |
| abstract_inverted_index.результаты | 281, 319 |
| abstract_inverted_index.содержащих | 297 |
| abstract_inverted_index.уникальном | 285 |
| abstract_inverted_index.установили, | 195 |
| abstract_inverted_index.абсолютного | 201 |
| abstract_inverted_index.взаимосвязь | 322 |
| abstract_inverted_index.связывающее | 312 |
| abstract_inverted_index.уточнениями | 215 |
| abstract_inverted_index.определения, | 228 |
| abstract_inverted_index.бесконечного | 310 |
| abstract_inverted_index.естественная | 231 |
| abstract_inverted_index.классические | 259 |
| abstract_inverted_index.обеспечивают | 247 |
| abstract_inverted_index.подчёркивают | 320 |
| abstract_inverted_index.последующими | 214 |
| abstract_inverted_index.комбинаторные | 192 |
| abstract_inverted_index.положительных | 234 |
| abstract_inverted_index.представлении | 286 |
| abstract_inverted_index.комбинаторики, | 326 |
| abstract_inverted_index.действительных | 287 |
| abstract_inverted_index.закономерности | 269 |
| abstract_inverted_index.приблизительно | 219 |
| abstract_inverted_index.рассматриваются | 179 |
| abstract_inverted_index.последовательности | 182, 186, 205 |
| abstract_inverted_index.последовательностей | 240 |
| abstract_inverted_index.последовательностей, | 332 |
| abstract_inverted_index.последовательностями, | 272 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |