The Seed-1 Power-Ceiling Orbit of Pi

Start with 1 and repeatedly multiply by pi, rounding up each time: 1, 4, 13, 41, 129, 406, 1276, 4009, 12595, 39569, 124310, ... This note gives a self-contained treatment of that seed-1 power-ceiling orbit. Iteration of the map x↦⌈sx⌉x \mapsto \lceil sx\rceil is classical, and Odlyzko and Wilf studied it with this exact seed in connection with the Josephus problem. The note specializes the theory to s=πs=\pi, while carefully distinguishing the seed-1 convention from the alternative power-ceiling convention that begins at ⌈s⌉\lceil s\rceil. The paper records the classical normalized asymptotics and the exact representation a(n)=⌊Cππn⌋a(n)=\lfloor C_\pi \pi^n\rfloor, together with two equivalent descriptions of the recurrence: a moving-window selection rule and a greedy lacunary characterization. It also gives the exact interval of multipliers compatible with any finite prefix. A general result shows that any sequence with finite, nonzero, transcendental exponential growth rate cannot be P-recursive over Q\mathbb{Q}, by a D-finite singularity argument. Since the present orbit grows at rate π\pi, the seed-1 π\pi orbit is therefore not P-recursive. The release also includes the first 1000 terms, high-precision generation code, interval-arithmetic verification, and a reproducibility warning: ordinary IEEE-754 double-precision arithmetic already produces an incorrect term at index 31.