Numerical identification of the error expansion of Brouncker's continued fraction for 4/π.

A short computational note on Brouncker's continued fraction 4/π = 1 + 1²/(2 + 3²/(2 + 5²/(2 + …))). With e_N the error after N partial quotients, the note determines numerically that (−1)^(N+1)·N·e_N tends to 4/π², and that the next coefficients of the expansion in 1/N are polynomials in 1/π whose odd-degree terms change sign with the parity of N. All four coefficients found so far are identified by PSLQ, and an independent computation (separately written code, 220-digit arithmetic) confirms them to between about 71 and 88 digits. Nothing here is proved. The note records the computation, states what was checked in the literature, and invites references to earlier sources. Files: Brouncker_error_expansion.pdf: the note (3 pages). Brouncker_error_expansion.tex: LaTeX source. brouncker_error_fit.py: Python/mpmath script that reproduces every number. brouncker_error_fit_output.txt: the script's full output. Made with AI assistance. Related: PCF Research Atlas (10.5281/zenodo.23317212, all versions).