{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:OD7YMH7TDHJAXMBSEIQC742J6M","short_pith_number":"pith:OD7YMH7T","canonical_record":{"source":{"id":"1611.08244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-24T16:39:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8e868cf99e8bf42b6e85a277418074b69140dcf37906a01e76ee6075a42e7962","abstract_canon_sha256":"2e3c8c230fba0ba39e6812cd73ab89c9cc0e7098bdfd19017869122e1d985fa1"},"schema_version":"1.0"},"canonical_sha256":"70ff861ff319d20bb03222202ff349f33861993380f8f05b4bba7239ae250406","source":{"kind":"arxiv","id":"1611.08244","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08244","created_at":"2026-05-18T00:56:39Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08244v1","created_at":"2026-05-18T00:56:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08244","created_at":"2026-05-18T00:56:39Z"},{"alias_kind":"pith_short_12","alias_value":"OD7YMH7TDHJA","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OD7YMH7TDHJAXMBS","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OD7YMH7T","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:OD7YMH7TDHJAXMBSEIQC742J6M","target":"record","payload":{"canonical_record":{"source":{"id":"1611.08244","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-24T16:39:08Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"8e868cf99e8bf42b6e85a277418074b69140dcf37906a01e76ee6075a42e7962","abstract_canon_sha256":"2e3c8c230fba0ba39e6812cd73ab89c9cc0e7098bdfd19017869122e1d985fa1"},"schema_version":"1.0"},"canonical_sha256":"70ff861ff319d20bb03222202ff349f33861993380f8f05b4bba7239ae250406","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:56:39.760171Z","signature_b64":"+8GnjM1o/Yb6/ueiNQ9cy0znUe/TkISIDFedAkZZu7TH7Ip5QCEQLRwHQPM5VQftaM60kaqEmWqsB9C4HgYIAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"70ff861ff319d20bb03222202ff349f33861993380f8f05b4bba7239ae250406","last_reissued_at":"2026-05-18T00:56:39.759376Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:56:39.759376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1611.08244","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":"2026-05-18T00:56:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tT4UkFamFAIpBIrlTXl4brl4wT8fgo1vuNcXwqEC9A2ibk232jdgJFH5yULsD9JXCOXWhIhBdjjWKC3W2HOhBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-21T04:34:51.659308Z"},"content_sha256":"0030447eae0af3def17e3357474a83de84bf7893a8888635dc0eb9879809f5ed","schema_version":"1.0","event_id":"sha256:0030447eae0af3def17e3357474a83de84bf7893a8888635dc0eb9879809f5ed"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:OD7YMH7TDHJAXMBSEIQC742J6M","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A Slow Relative of Hofstadter's Q-Sequence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Nathan Fox","submitted_at":"2016-11-24T16:39:08Z","abstract_excerpt":"Hofstadter's Q-sequence remains an enigma fifty years after its introduction. Initially, the terms of the sequence increase monotonically by 0 or 1 at a time. But, Q(12)=8 while Q(11)=6, and monotonicity fails shortly thereafter. In this paper, we add a third term to Hofstadter's recurrence, giving the recurrence B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3)). We show that this recurrence, along with a suitable initial condition that naturally generalizes Hofstadter's initial condition, generates a sequence whose terms all increase monotonically by 0 or 1 at a time. Furthermore, we give a complete d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08244","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":"2026-05-18T00:56:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wrA2aRfCiJQrCSSsUi5/D+z4M5/ZEtluWGy8iD+cUTTs0WOHKqMTFWK8RhiU8PkqMAwq1r4tR1+VOy9WdoOMDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-21T04:34:51.659675Z"},"content_sha256":"21129c0716634099a64678158aa216e3ebea305ea0024d3911663cd29cf4020c","schema_version":"1.0","event_id":"sha256:21129c0716634099a64678158aa216e3ebea305ea0024d3911663cd29cf4020c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/OD7YMH7TDHJAXMBSEIQC742J6M/bundle.json","state_url":"https://pith.science/pith/OD7YMH7TDHJAXMBSEIQC742J6M/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/OD7YMH7TDHJAXMBSEIQC742J6M/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":"2026-07-21T04:34:51Z","links":{"resolver":"https://pith.science/pith/OD7YMH7TDHJAXMBSEIQC742J6M","bundle":"https://pith.science/pith/OD7YMH7TDHJAXMBSEIQC742J6M/bundle.json","state":"https://pith.science/pith/OD7YMH7TDHJAXMBSEIQC742J6M/state.json","well_known_bundle":"https://pith.science/.well-known/pith/OD7YMH7TDHJAXMBSEIQC742J6M/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:OD7YMH7TDHJAXMBSEIQC742J6M","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":"2e3c8c230fba0ba39e6812cd73ab89c9cc0e7098bdfd19017869122e1d985fa1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-24T16:39:08Z","title_canon_sha256":"8e868cf99e8bf42b6e85a277418074b69140dcf37906a01e76ee6075a42e7962"},"schema_version":"1.0","source":{"id":"1611.08244","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08244","created_at":"2026-05-18T00:56:39Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08244v1","created_at":"2026-05-18T00:56:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08244","created_at":"2026-05-18T00:56:39Z"},{"alias_kind":"pith_short_12","alias_value":"OD7YMH7TDHJA","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"OD7YMH7TDHJAXMBS","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"OD7YMH7T","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:21129c0716634099a64678158aa216e3ebea305ea0024d3911663cd29cf4020c","target":"graph","created_at":"2026-05-18T00:56:39Z","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":"Hofstadter's Q-sequence remains an enigma fifty years after its introduction. Initially, the terms of the sequence increase monotonically by 0 or 1 at a time. But, Q(12)=8 while Q(11)=6, and monotonicity fails shortly thereafter. In this paper, we add a third term to Hofstadter's recurrence, giving the recurrence B(n)=B(n-B(n-1))+B(n-B(n-2))+B(n-B(n-3)). We show that this recurrence, along with a suitable initial condition that naturally generalizes Hofstadter's initial condition, generates a sequence whose terms all increase monotonically by 0 or 1 at a time. Furthermore, we give a complete d","authors_text":"Nathan Fox","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-24T16:39:08Z","title":"A Slow Relative of Hofstadter's Q-Sequence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08244","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":"sha256:0030447eae0af3def17e3357474a83de84bf7893a8888635dc0eb9879809f5ed","target":"record","created_at":"2026-05-18T00:56:39Z","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":"2e3c8c230fba0ba39e6812cd73ab89c9cc0e7098bdfd19017869122e1d985fa1","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-11-24T16:39:08Z","title_canon_sha256":"8e868cf99e8bf42b6e85a277418074b69140dcf37906a01e76ee6075a42e7962"},"schema_version":"1.0","source":{"id":"1611.08244","kind":"arxiv","version":1}},"canonical_sha256":"70ff861ff319d20bb03222202ff349f33861993380f8f05b4bba7239ae250406","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"70ff861ff319d20bb03222202ff349f33861993380f8f05b4bba7239ae250406","first_computed_at":"2026-05-18T00:56:39.759376Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:56:39.759376Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+8GnjM1o/Yb6/ueiNQ9cy0znUe/TkISIDFedAkZZu7TH7Ip5QCEQLRwHQPM5VQftaM60kaqEmWqsB9C4HgYIAA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:56:39.760171Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.08244","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0030447eae0af3def17e3357474a83de84bf7893a8888635dc0eb9879809f5ed","sha256:21129c0716634099a64678158aa216e3ebea305ea0024d3911663cd29cf4020c"],"state_sha256":"4197db260c63fac68941664b03a3c1bc9d51007679efc244dd2a2f03a51782fe"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"hBsze9QXWp3YAZTWRaicEA/pkrxw8gPqmWshJgR3yRYR1Xlf77pScmPAm4nTPEk+lKLrJ2svhnp6lgoIhA/yAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-21T04:34:51.662328Z","bundle_sha256":"282ac2ac20fd9849574bb5f1ebef6337542c05fdcadf4ff77f9c0027a8f97549"}}