A New Symmetrical Numeric Pattern with Analytical Features Based on Palindromic Numbers and the Palindrome Diamond

In the realms of discrete mathematics, number theory, and algorithm analyses, palindromes, which are numbers that read the same backwards as forwards, have long been a subject of interest. Although earlier research was dominated by palindromic primes, symmetry of digits, and ways of calculation to obtain palindromes, structured presentations of palindromic sequences in multi-dimensional plots have yet to be completely investigated. In this paper, we discuss and mathematically analyze the newly proposed numerical structure known as the Palindrome Diamond. Palindrome Diamond is a symmetrical diamond structure consisting of palindromic numbers arranged in a diamond pattern; this pattern is determined by the central number n. In Palindrome Diamond, each row consists of an palindromic pattern that gradually approaches the center and then retreats symmetrically towards the sides. The proposed structure is fully symmetrical in the horizontal and vertical planes. Further formulae are also provided to determine the overall quantity of numbers and summation of all Palindrome Diamonds based on a specific center number n. Applications involving the Palindrome Diamond include mathematical visualization, algorithm design, and teaching discrete mathematics via connections between geometric symmetry, numerical palindromes, and combinatorial pattern generation.