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   A321305 Triangle T(n,f): the number of signed cubic graphs on 2n vertices with f edges of the first sign. 2
   1, 0, 0, 0, 0, 1, 1, 2, 3, 2, 1, 1, 2, 3, 8, 16, 21, 21, 16, 8, 3, 2, 5, 14, 57, 152, 313, 474, 551, 474, 313, 152, 57, 14, 5, 19, 91, 491, 1806, 5034, 10604, 17318, 22033, 22033, 17318, 10604, 5034, 1806, 491, 91, 19, 85, 706, 4981, 23791, 84575, 229078, 487020, 825127, 1127783, 1250632, 1127783, 825127, 487020, 229078, 84575, 23791, 4981, 706, 85 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,8 COMMENTS These are connected, undirected, simple cubic graphs where each edge is signed as either "+" or "-". Row n has 1+3n entries, 0<=f<=3n. The column f=0 (1, 0, 1, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 3, 14, 91, 706,...) counts the edge-rooted cubic graphs. LINKS FORMULA T(n,f) = T(n,3*n-f). EXAMPLE The triangle starts: 0 vertices: 1 2 vertices: 0,0,0,0 4 vertices: 1,1,2,3,2,1,1 6 vertices: 2,3,8,16,21,21,16,8,3,2 8 vertices: 5,14,57,152,313,474,551,474,313,152,57,14,5 10 vertices: 19,91,491,1806,5034,10604,17318,22033,22033,17318,10604,5034,1806,491,91,19 CROSSREFS Cf. A002851 (first column), A321304 (signed vertices), A302939 (signed trees). Sequence in context: A105734 A076839 A092542 * A339178 A026552 A333271 Adjacent sequences:  A321302 A321303 A321304 * A321306 A321307 A321308 KEYWORD nonn,tabf AUTHOR R. J. Mathar, Nov 03 2018 STATUS approved
  
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  Last modified August 14 17:12 EDT 2022. Contains 356122 sequences. (Running on oeis4.)
explanation: The provided text is an excerpt from the Online Encyclopedia of Integer Sequences (OEIS) describing the sequence A321305. It details a triangle T(n,f) that represents the number of signed cubic graphs on 2n vertices with f edges of the first sign. The text includes the definition, comments, a formula, and an example of the triangle. It also provides cross-references to related sequences and metadata about the sequence's approval status.
  
  The document does not contain a specific math question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: The OEIS is supported by the many generous donors to the OEIS Foundation.
  
   Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
   A321305 Triangle T(n,f): the number of signed cubic graphs on 2n vertices with f edges of the first sign. 2
   1, 0, 0, 0, 0, 1, 1, 2, 3, 2, 1, 1, 2, 3, 8, 16, 21, 21, 16, 8, 3, 2, 5, 14, 57, 152, 313, 474, 551, 474, 313, 152, 57, 14, 5, 19, 91, 491, 1806, 5034, 10604, 17318, 22033, 22033, 17318, 10604, 5034, 1806, 491, 91, 19, 85, 706, 4981, 23791, 84575, 229078, 487020, 825127, 1127783, 1250632, 1127783, 825127, 487020, 229078, 84575, 23791, 4981, 706, 85 (list; graph; refs; listen; history; text; internal format)
   OFFSET 0,8 COMMENTS These are connected, undirected, simple cubic graphs where each edge is signed as either "+" or "-". Row n has 1+3n entries, 0<=f<=3n. The column f=0 (1, 0, 1, 2, 5,...) counts the cubic graphs (A002851). The column f=1 (0, 1, 3, 14, 91, 706,...) counts the edge-rooted cubic graphs. LINKS FORMULA T(n,f) = T(n,3*n-f). EXAMPLE The triangle starts: 0 vertices: 1 2 vertices: 0,0,0,0 4 vertices: 1,1,2,3,2,1,1 6 vertices: 2,3,8,16,21,21,16,8,3,2 8 vertices: 5,14,57,152,313,474,551,474,313,152,57,14,5 10 vertices: 19,91,491,1806,5034,10604,17318,22033,22033,17318,10604,5034,1806,491,91,19 CROSSREFS Cf. A002851 (first column), A321304 (signed vertices), A302939 (signed trees). Sequence in context: A105734 A076839 A092542 * A339178 A026552 A333271 Adjacent sequences:  A321302 A321303 A321304 * A321306 A321307 A321308 KEYWORD nonn,tabf AUTHOR R. J. Mathar, Nov 03 2018 STATUS approved
  
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  Last modified August 14 17:12 EDT 2022. Contains 356122 sequences. (Running on oeis4.)
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is an excerpt from the Online Encyclopedia of Integer Sequences (OEIS) describing the sequence A321305. It details a triangle T(n,f) that represents the number of signed cubic graphs on 2n vertices with f edges of the first sign. The text includes the definition, comments, a formula, and an example of the triangle. It also provides cross-references to related sequences and metadata about the sequence's approval status.
  
  The document does not contain a specific math question or a question and answer.
  
  Label: no-question