Palindromes, Pascal Triangles, and Digitals Convolutions in Base b
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| Formato: | Recurso digital |
| Lenguaje: | inglés |
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2025
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| _version_ | 1866901265984782336 |
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| author | Coronato, Antonio |
| author_facet | Coronato, Antonio |
| contents | <p>We study even palindromes in base b through a new viewpoint based on discrete convolutions and Pascal-type transforms. Given an integer N, the associated palindrome T_b(N) can be interpreted as a digital autocorrelation of its base-b digits, while division by b+1 corresponds to a binomial transform perturbed by localized carry effects. We introduce the ideal quotient, obtained by enforcing symmetric carries, and show that its digits arise from a folded Pascal-triangle structure. Numerical experiments across several bases reveal unexpected regularity phenomena, including high palindromicity rates in the diagonal products U(N,N)=T_b(N)^2/(b+1) and universal “smooth” digit patterns. We formulate classification and probabilistic questions motivated by these findings.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17716690 |
| institution | Zenodo |
| language | eng |
| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Palindromes, Pascal Triangles, and Digitals Convolutions in Base b Coronato, Antonio Palindromic numbers, Base-b numeration systems, Digit convolutions, Pascal triangles transforms, Binomial transforms, Cayy propagation, Borrow chains, Palindrome quotients, Divisibility by b+1, Digital autocorretion, Symmetric digit patterns. <p>We study even palindromes in base b through a new viewpoint based on discrete convolutions and Pascal-type transforms. Given an integer N, the associated palindrome T_b(N) can be interpreted as a digital autocorrelation of its base-b digits, while division by b+1 corresponds to a binomial transform perturbed by localized carry effects. We introduce the ideal quotient, obtained by enforcing symmetric carries, and show that its digits arise from a folded Pascal-triangle structure. Numerical experiments across several bases reveal unexpected regularity phenomena, including high palindromicity rates in the diagonal products U(N,N)=T_b(N)^2/(b+1) and universal “smooth” digit patterns. We formulate classification and probabilistic questions motivated by these findings.</p> |
| title | Palindromes, Pascal Triangles, and Digitals Convolutions in Base b |
| topic | Palindromic numbers, Base-b numeration systems, Digit convolutions, Pascal triangles transforms, Binomial transforms, Cayy propagation, Borrow chains, Palindrome quotients, Divisibility by b+1, Digital autocorretion, Symmetric digit patterns. |
| url | https://doi.org/10.5281/zenodo.17716690 |