{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:CKGDW6WXLMVEGXPCWKBW4LSV22","short_pith_number":"pith:CKGDW6WX","canonical_record":{"source":{"id":"1501.00061","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-31T01:28:12Z","cross_cats_sorted":[],"title_canon_sha256":"31376d1e5e38e292224f2f5d35f847c030a272f85c1c650efd61c1804bc743b2","abstract_canon_sha256":"a0b0700517db9166047d97c3f9159667441acb0ad943f2132d7fd10e9dde37bf"},"schema_version":"1.0"},"canonical_sha256":"128c3b7ad75b2a435de2b2836e2e55d6a10dd925d394c27245449dc523ca1efe","source":{"kind":"arxiv","id":"1501.00061","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00061","created_at":"2026-05-18T02:30:10Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00061v1","created_at":"2026-05-18T02:30:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00061","created_at":"2026-05-18T02:30:10Z"},{"alias_kind":"pith_short_12","alias_value":"CKGDW6WXLMVE","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CKGDW6WXLMVEGXPC","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CKGDW6WX","created_at":"2026-05-18T12:28:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:CKGDW6WXLMVEGXPCWKBW4LSV22","target":"record","payload":{"canonical_record":{"source":{"id":"1501.00061","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-31T01:28:12Z","cross_cats_sorted":[],"title_canon_sha256":"31376d1e5e38e292224f2f5d35f847c030a272f85c1c650efd61c1804bc743b2","abstract_canon_sha256":"a0b0700517db9166047d97c3f9159667441acb0ad943f2132d7fd10e9dde37bf"},"schema_version":"1.0"},"canonical_sha256":"128c3b7ad75b2a435de2b2836e2e55d6a10dd925d394c27245449dc523ca1efe","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:30:10.264438Z","signature_b64":"/uv4wiorX9Nkz1P/6HTFyi/Z0YUTCA7wpSnKPz+Ra2fUuq3PNWp7mO19NCCUyJMPbKmATZxwkb4Nsma7c7yKDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"128c3b7ad75b2a435de2b2836e2e55d6a10dd925d394c27245449dc523ca1efe","last_reissued_at":"2026-05-18T02:30:10.264006Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:30:10.264006Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.00061","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:30:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"r7PO8Fc+9+Y9RanHQbSMrkFY0OO35AusxPo6P7eOBRB/WqBdX0LzeLe0N/I3QZADT7JGUu8RFzLI5iLhactzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:58:56.077763Z"},"content_sha256":"7c43446c8be1291c25eb1605616d17d0bd4675fc80eea492787ef1987381dabd","schema_version":"1.0","event_id":"sha256:7c43446c8be1291c25eb1605616d17d0bd4675fc80eea492787ef1987381dabd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:CKGDW6WXLMVEGXPCWKBW4LSV22","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A graph theoretic encoding of Lucas sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"James Alexander, Paul Hearding","submitted_at":"2014-12-31T01:28:12Z","abstract_excerpt":"Some well-known results of Prodinger and Tichy are that the number of independent sets in the $n$-vertex path graph is $F_{n+2}$, and that the number of independent sets in the $n$-vertex cycle graph is $L_n$. We generalize these results by introducing new classes of graphs whose independent set structures encode the Lucas sequences of both the first and second kind. We then use this class of graphs to provide new combinatorial interpretations of the terms of Dickson polynomials of the first and second kind."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00061","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:30:10Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"93yFd6gKGbCSm6Ht01p827W+lB+thWEQscrfcSV/xoZzAPZwWRYnXliSPjP5xcM/InabfWUnnj46U7a+0nqQDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-25T11:58:56.078410Z"},"content_sha256":"61907415ef63512df7c539b02597fab569924b5445bd1a1c7b39ec5edc88f7fc","schema_version":"1.0","event_id":"sha256:61907415ef63512df7c539b02597fab569924b5445bd1a1c7b39ec5edc88f7fc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CKGDW6WXLMVEGXPCWKBW4LSV22/bundle.json","state_url":"https://pith.science/pith/CKGDW6WXLMVEGXPCWKBW4LSV22/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CKGDW6WXLMVEGXPCWKBW4LSV22/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-25T11:58:56Z","links":{"resolver":"https://pith.science/pith/CKGDW6WXLMVEGXPCWKBW4LSV22","bundle":"https://pith.science/pith/CKGDW6WXLMVEGXPCWKBW4LSV22/bundle.json","state":"https://pith.science/pith/CKGDW6WXLMVEGXPCWKBW4LSV22/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CKGDW6WXLMVEGXPCWKBW4LSV22/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:CKGDW6WXLMVEGXPCWKBW4LSV22","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"a0b0700517db9166047d97c3f9159667441acb0ad943f2132d7fd10e9dde37bf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-31T01:28:12Z","title_canon_sha256":"31376d1e5e38e292224f2f5d35f847c030a272f85c1c650efd61c1804bc743b2"},"schema_version":"1.0","source":{"id":"1501.00061","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.00061","created_at":"2026-05-18T02:30:10Z"},{"alias_kind":"arxiv_version","alias_value":"1501.00061v1","created_at":"2026-05-18T02:30:10Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.00061","created_at":"2026-05-18T02:30:10Z"},{"alias_kind":"pith_short_12","alias_value":"CKGDW6WXLMVE","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"CKGDW6WXLMVEGXPC","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"CKGDW6WX","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:61907415ef63512df7c539b02597fab569924b5445bd1a1c7b39ec5edc88f7fc","target":"graph","created_at":"2026-05-18T02:30:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Some well-known results of Prodinger and Tichy are that the number of independent sets in the $n$-vertex path graph is $F_{n+2}$, and that the number of independent sets in the $n$-vertex cycle graph is $L_n$. We generalize these results by introducing new classes of graphs whose independent set structures encode the Lucas sequences of both the first and second kind. We then use this class of graphs to provide new combinatorial interpretations of the terms of Dickson polynomials of the first and second kind.","authors_text":"James Alexander, Paul Hearding","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-31T01:28:12Z","title":"A graph theoretic encoding of Lucas sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.00061","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7c43446c8be1291c25eb1605616d17d0bd4675fc80eea492787ef1987381dabd","target":"record","created_at":"2026-05-18T02:30:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"a0b0700517db9166047d97c3f9159667441acb0ad943f2132d7fd10e9dde37bf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-12-31T01:28:12Z","title_canon_sha256":"31376d1e5e38e292224f2f5d35f847c030a272f85c1c650efd61c1804bc743b2"},"schema_version":"1.0","source":{"id":"1501.00061","kind":"arxiv","version":1}},"canonical_sha256":"128c3b7ad75b2a435de2b2836e2e55d6a10dd925d394c27245449dc523ca1efe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"128c3b7ad75b2a435de2b2836e2e55d6a10dd925d394c27245449dc523ca1efe","first_computed_at":"2026-05-18T02:30:10.264006Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:30:10.264006Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/uv4wiorX9Nkz1P/6HTFyi/Z0YUTCA7wpSnKPz+Ra2fUuq3PNWp7mO19NCCUyJMPbKmATZxwkb4Nsma7c7yKDg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:30:10.264438Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.00061","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c43446c8be1291c25eb1605616d17d0bd4675fc80eea492787ef1987381dabd","sha256:61907415ef63512df7c539b02597fab569924b5445bd1a1c7b39ec5edc88f7fc"],"state_sha256":"2a6006183d1976290df34cd16d55b87ac59d684295b6cd088edcce55f485ade5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rmWYhWaDx9mOBGFTaL9+K89adml8tUOcSOzMlegnOJip/PPBLqteJgcILkJEy8gYu1dzl2EnBWiad0Rkpm5+DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-25T11:58:56.081603Z","bundle_sha256":"87b1df25c0c4db2785c485af2c2306f2b20ed1ab4c234a63ebe5538cb7c7e32c"}}