AI 中文总结
该研究为OEIS中猜想提供进一步证明，涉及初等与乘法数论等多领域问题，运用组合双射、同余等多种方法，部分结果给出序列完整特征，部分建立恒等式等，推动了相关数学领域的研究。
AI 中文摘要
这是致力于证明整数序列在线百科全书（OEIS）中所记录猜想的系列工作的第四篇。所考虑的问题涉及初等和乘法数论、斐波那契数、十进制连接、丢番图方程和佩尔方程、二进制表示和按位运算、格路径、奇偶模式、递归以及形式幂级数。部分结果给出了相关序列的完整特征；其他结果建立了精确的恒等式、递归、生成函数、渐近估计、整数性性质或不存在性结果。证明使用了组合双射、同余、莫比乌斯反演、赋值、斐波那契恒等式、佩尔型论证、卢卡斯定理、里奥丹阵列、拉格朗日反演和生成函数方法。
英文摘要
This is the fourth work in a series devoted to proving conjectures recorded in the On-Line Encyclopedia of Integer Sequences (OEIS). The problems considered here concern elementary and multiplicative number theory, Fibonacci numbers, decimal concatenation, Diophantine and Pell equations, binary representations and bitwise operations, lattice paths, parity patterns, recurrences, and formal power series. Several of the results give complete characterizations of the relevant sequences; others establish exact identities, recurrences, generating functions, asymptotic estimates, integrality properties, or nonoccurrence results. The proofs use combinatorial bijections, congruences, Möbius inversion, valuations, Fibonacci identities, Pell-type arguments, Lucas' theorem, Riordan arrays, Lagrange inversion, and generating-function methods.
Comments57 pages
