{"paper":{"title":"A method for deterining the mod-$3^k$ behaviour of recursive sequences","arxiv_id":"1308.2856","primary_cat":"math.CO","submitted_at":"2013-08-13T13:19:19+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13-20260917 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":0.75,"id":"s0","own":true,"role":"contribution","text":"We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.36,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.25,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.75,"interpretation":0.0}},{"i":1,"p":0.85,"id":"s1","own":true,"role":"contribution","text":"We apply this method to Catalan numbers, Motzkin numbers, Riordan numbers, Schr\\\"oder numbers, Eulerian numbers, trinomial coefficients, Delannoy numbers, and to functions counting free subgroups of finite index in the inhomogeneous modular group and its lifts.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.92,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.14,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.85,"interpretation":0.0}},{"i":2,"p":0.86,"id":"s2","own":true,"role":"result","text":"This leads to numerous new results, including many extensions of known results to higher powers of 3.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.9,"main_claim_p":0.45,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.86,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.07,"interpretation":0.07}}]},"compiled":{"main_claim":{"p":0.84,"text":"This leads to numerous new results, including many extensions of known results to higher powers of 3.","location":"abstract s2","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"OpenRouter","model":"classifier-1.13","endpoint":"https://openrouter.ai/api/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":2990,"cost_usd_reported":0.000126},"compiled_at":"2026-09-23","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13-20260917","abstract_states":{"claims_first":0.04,"states_limitation":0.09,"human_participants":0.01,"controlled_comparison":0.01,"randomized_assignment":0.01,"states_quantitative_result":0.07,"compares_with_prior_results":0.42,"mentions_code_or_data_availability":0.02},"evidential_basis":{"p":1.0,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":1.0,"analysis_of_existing_data":0.0,"observation_or_experiment":0.0,"simulation_or_computation":0.0}},"contribution_type":{"p":1.0,"value":"method or tool","probabilities":{"review":0.0,"method_or_tool":1.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":0.54,"text":"We present a method for obtaining congruences modulo powers of 3 for sequences given by recurrences of finite depth with polynomial coefficients.","location":"abstract s0"}},"coverage":{"tier":"abstract","levels_run":["P abstract"],"not_examined":["full text","figures and tables","references and citing sentences","relations to other papers"],"sources_read":["arXiv listing abstract v1"],"levels_not_run":["S sections","T body sentences","C citations","R cross-paper relations"],"sentences_examined":3,"sentences_in_abstract":3},"measured_accuracy":{"url":"https://pith.science/accuracy","measured_on":"2026-10-02","method":"two independent model families labelled a fresh random sample blind; the figure is how often the published label matches their consensus","sentence_role_agreement":0.92,"main_claim_agreement":0.908,"typed_citation_agreement":0.933,"typed_citation_agreement_95_interval":[0.917,0.945],"review_comments_accurate":0.758,"review_comments_accurate_95_interval":[0.684,0.82],"review_caught_admitted_error_found_or_related":0.448}}