Calkin–Wilf tree (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Calkin–Wilf tree" in English language version.

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Carlitz (1964). Carlitz, L. (1964), "A problem in partitions related to the Stirling numbers", Bulletin of the American Mathematical Society, 70 (2): 275–278, doi:10.1090/S0002-9904-1964-11118-6, MR 0157907.
The OEIS entry credits this fact to Carlitz (1964) and to uncited work of Lind. However, Carlitz's paper describes a more restricted class of sums of powers of two, counted by $fusc(n)$ instead of by $fusc(n + 1)$ . Carlitz, L. (1964), "A problem in partitions related to the Stirling numbers", Bulletin of the American Mathematical Society, 70 (2): 275–278, doi:10.1090/S0002-9904-1964-11118-6, MR 0157907.
Bates, Bunder & Tognetti (2010). Bates, Bruce; Bunder, Martin; Tognetti, Keith (2010), "Linking the Calkin-Wilf and Stern-Brocot trees", European Journal of Combinatorics, 31 (7): 1637–1661, doi:10.1016/j.ejc.2010.04.002, MR 2673006

archive.org

Kepler, J. (1619), Harmonices Mundi, vol. III, p. 27.

doi.org

Berstel & de Luca (1997), Section 6. Berstel, Jean; de Luca, Aldo (1997), "Sturmian words, Lyndon words and trees", Theoretical Computer Science, 178 (1–2): 171–203, doi:10.1016/S0304-3975(96)00101-6
Raney (1973). Raney, George N. (1973), "On continued fractions and finite automata", Mathematische Annalen, 206 (4): 265–283, doi:10.1007/BF01355980, hdl:10338.dmlcz/128216, S2CID 120933574
Gibbons, Lester & Bird (2006). Gibbons, Jeremy; Lester, David; Bird, Richard (2006), "Functional pearl: Enumerating the rationals", Journal of Functional Programming, 16 (3): 281–291, doi:10.1017/S0956796806005880 (inactive 6 January 2025), S2CID 14237968{{citation}}: CS1 maint: DOI inactive as of January 2025 (link).
Calkin & Wilf (2000): "a list of all positive rational numbers, each appearing once and only once, can be made by writing down ⁠1/1⁠, then the fractions on the level just below the top of the tree, reading from left to right, then the fractions on the next level down, reading from left to right, etc." Gibbons, Lester & Bird (2006) discuss efficient functional programming techniques for performing this breadth first traversal. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182. Gibbons, Jeremy; Lester, David; Bird, Richard (2006), "Functional pearl: Enumerating the rationals", Journal of Functional Programming, 16 (3): 281–291, doi:10.1017/S0956796806005880 (inactive 6 January 2025), S2CID 14237968{{citation}}: CS1 maint: DOI inactive as of January 2025 (link).
Knuth et al. (2003) credit this formula to Moshe Newman. Knuth, Donald E.; Rupert, C.P.; Smith, Alex; Stong, Richard (2003), "Recounting the Rationals, Continued: 10906", The American Mathematical Monthly, 110 (7): 642–643, doi:10.2307/3647762, JSTOR 3647762
Calkin & Wilf (2000), Theorem 1. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182.
Carlitz (1964). Carlitz, L. (1964), "A problem in partitions related to the Stirling numbers", Bulletin of the American Mathematical Society, 70 (2): 275–278, doi:10.1090/S0002-9904-1964-11118-6, MR 0157907.
The OEIS entry credits this fact to Carlitz (1964) and to uncited work of Lind. However, Carlitz's paper describes a more restricted class of sums of powers of two, counted by $fusc(n)$ instead of by $fusc(n + 1)$ . Carlitz, L. (1964), "A problem in partitions related to the Stirling numbers", Bulletin of the American Mathematical Society, 70 (2): 275–278, doi:10.1090/S0002-9904-1964-11118-6, MR 0157907.
Bates, Bunder & Tognetti (2010). Bates, Bruce; Bunder, Martin; Tognetti, Keith (2010), "Linking the Calkin-Wilf and Stern-Brocot trees", European Journal of Combinatorics, 31 (7): 1637–1661, doi:10.1016/j.ejc.2010.04.002, MR 2673006

handle.net

hdl.handle.net

Raney (1973). Raney, George N. (1973), "On continued fractions and finite automata", Mathematische Annalen, 206 (4): 265–283, doi:10.1007/BF01355980, hdl:10338.dmlcz/128216, S2CID 120933574

jstor.org

Calkin & Wilf (2000): "a list of all positive rational numbers, each appearing once and only once, can be made by writing down ⁠1/1⁠, then the fractions on the level just below the top of the tree, reading from left to right, then the fractions on the next level down, reading from left to right, etc." Gibbons, Lester & Bird (2006) discuss efficient functional programming techniques for performing this breadth first traversal. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182. Gibbons, Jeremy; Lester, David; Bird, Richard (2006), "Functional pearl: Enumerating the rationals", Journal of Functional Programming, 16 (3): 281–291, doi:10.1017/S0956796806005880 (inactive 6 January 2025), S2CID 14237968{{citation}}: CS1 maint: DOI inactive as of January 2025 (link).
Knuth et al. (2003) credit this formula to Moshe Newman. Knuth, Donald E.; Rupert, C.P.; Smith, Alex; Stong, Richard (2003), "Recounting the Rationals, Continued: 10906", The American Mathematical Monthly, 110 (7): 642–643, doi:10.2307/3647762, JSTOR 3647762
Calkin & Wilf (2000), Theorem 1. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182.

semanticscholar.org

api.semanticscholar.org

Raney (1973). Raney, George N. (1973), "On continued fractions and finite automata", Mathematische Annalen, 206 (4): 265–283, doi:10.1007/BF01355980, hdl:10338.dmlcz/128216, S2CID 120933574
Gibbons, Lester & Bird (2006). Gibbons, Jeremy; Lester, David; Bird, Richard (2006), "Functional pearl: Enumerating the rationals", Journal of Functional Programming, 16 (3): 281–291, doi:10.1017/S0956796806005880 (inactive 6 January 2025), S2CID 14237968{{citation}}: CS1 maint: DOI inactive as of January 2025 (link).
Calkin & Wilf (2000): "a list of all positive rational numbers, each appearing once and only once, can be made by writing down ⁠1/1⁠, then the fractions on the level just below the top of the tree, reading from left to right, then the fractions on the next level down, reading from left to right, etc." Gibbons, Lester & Bird (2006) discuss efficient functional programming techniques for performing this breadth first traversal. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182. Gibbons, Jeremy; Lester, David; Bird, Richard (2006), "Functional pearl: Enumerating the rationals", Journal of Functional Programming, 16 (3): 281–291, doi:10.1017/S0956796806005880 (inactive 6 January 2025), S2CID 14237968{{citation}}: CS1 maint: DOI inactive as of January 2025 (link).

uow.edu.au

ro.uow.edu.au

Bates, Bunder & Tognetti (2010). Bates, Bruce; Bunder, Martin; Tognetti, Keith (2010), "Linking the Calkin-Wilf and Stern-Brocot trees", European Journal of Combinatorics, 31 (7): 1637–1661, doi:10.1016/j.ejc.2010.04.002, MR 2673006

upenn.edu

math.upenn.edu

Calkin & Wilf (2000): "a list of all positive rational numbers, each appearing once and only once, can be made by writing down ⁠1/1⁠, then the fractions on the level just below the top of the tree, reading from left to right, then the fractions on the next level down, reading from left to right, etc." Gibbons, Lester & Bird (2006) discuss efficient functional programming techniques for performing this breadth first traversal. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182. Gibbons, Jeremy; Lester, David; Bird, Richard (2006), "Functional pearl: Enumerating the rationals", Journal of Functional Programming, 16 (3): 281–291, doi:10.1017/S0956796806005880 (inactive 6 January 2025), S2CID 14237968{{citation}}: CS1 maint: DOI inactive as of January 2025 (link).
Calkin & Wilf (2000), Theorem 1. Calkin, Neil; Wilf, Herbert (2000), "Recounting the rationals" (PDF), American Mathematical Monthly, 107 (4), Mathematical Association of America: 360–363, doi:10.2307/2589182, JSTOR 2589182.