Index recommendation is essential for improving query performance in database management systems (DBMSs) through creating an optimal set of indexes under specific constraints. Traditional methods, such as heuristic and learning-based approaches, are effective but face challenges like lengthy recommendation time, resource-intensive training, and poor generalization across different workloads and database schemas. To address these issues, we propose LLMIdxAdvis, a resource-efficient index advisor that uses large language models (LLMs) without extensive fine-tuning. LLMIdxAdvis frames index recommendation as a sequence-to-sequence task, taking target workload, storage constraint, and corresponding database environment as input, and directly outputting recommended indexes. It constructs a high-quality demonstration pool offline, using GPT-4-Turbo to synthesize diverse SQL queries and applying integrated heuristic methods to collect both default and refined labels. During recommendation, these demonstrations are ranked to inject database expertise via in-context learning. Additionally, LLMIdxAdvis extracts workload features involving specific column statistical information to strengthen LLM's understanding, and introduces a novel inference scaling strategy combining vertical scaling (via ''Index-Guided Major Voting'' and Best-of-N) and horizontal scaling (through iterative ''self-optimization'' with database feedback) to enhance reliability. Experiments on 3 OLAP and 2 real-world benchmarks reveal that LLMIdxAdvis delivers competitive index recommendation with reduced runtime, and generalizes effectively across different workloads and database schemas.