{"id":"W10043420","doi":"10.1103/physrevlett.66.2193","title":"Catalan Numbers, Lucas Numbers, and Circuits.","year":2005,"lang":"en","type":"article","venue":"Ars Combinatoria","topic":"Advanced Mathematical Theories and Applications","field":"Physics and Astronomy","cited_by":0,"is_retracted":false,"has_abstract":false,"ca_institutions":"","funders":"","keywords":"Mathematics; Catalan number; Lucas number; Arithmetic; Lucas sequence; Combinatorics; Fibonacci number; Fibonacci polynomials","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"codex-gemma-dda1882f352a","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.00005996555,0.0001158961,0.0001416187,0.00001250154,0.0001412233,0.00003549841,0.0001075947,0.0000257245,0.0003041393],"category_scores_gemma":[0.000005626131,0.0001117468,0.00003990432,0.00009529472,0.00007237658,0.0001262408,0.00005588685,0.0001046668,0.0001747821],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00001563137,"about_ca_system_score_gemma":0.00001316411,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.00004061911,"about_ca_topic_score_gemma":0.000001569804,"domain_scores_codex":[0.9993764,0.00001038937,0.0001472141,0.0001848657,0.00008278686,0.0001983789],"domain_scores_gemma":[0.9994936,0.00007400718,0.00005263212,0.0002406605,0.00002710202,0.0001119729],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[8.566702e-7,0.00007994562,0.001683202,0.000004543346,0.0000162484,2.178383e-7,0.000196947,0.000001668741,0.00008571788,0.9864541,0.001351685,0.01012481],"study_design_scores_gemma":[0.00035209,0.000009549368,0.0002465913,0.000008400619,0.0000142961,0.000001667931,0.0002356923,0.0000372179,0.0007760405,0.9493081,0.04886553,0.0001448475],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.8134575,0.0001276479,0.003060692,0.001100747,0.0002794798,0.0003439492,0.00004511588,0.0001152299,0.1814696],"genre_scores_gemma":[0.9975863,0.000004542094,0.0009696882,0.00005923042,0.00004663362,0.00003764697,0.00002377229,0.00001667961,0.001255506],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.1841288,"threshold_uncertainty_score":0.4556902,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.007480950295347841,"score_gpt":0.2529497069257358,"score_spread":0.2454687566303879,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). Scores rank; they never assert a category."}}