Clark Kimberling

Professor of Mathematics, University of Evansville, 1800 Lincoln Avenue, Evansville, IN 47722
Welcome to my home page. Here you'll find various mathematical, historical, and musical topics, in many cases matched with biographical sketches. The mathematical topics include triangle geometry and number theory. The section on triangle geometry contains the world's largest internet collection of triangle centers. The number-theoretic topics deal mostly with integer sequences and arrays.

The historical collection started out with 19th century scientists associated with the famous town, New Harmony, Indiana, located about 30 miles from Evansville. The collection has grown to include educators, writers, and artists.

The music section consists of projects, including links to free downloadable scores and recordings. Ongoing projects include arranging and composing for groups of recorders. Below, you can click and listen to 75 duets and 117 trios performed by Japanese recorder virtuouso Papalin.


Triangle Centers: Classical and Recent
Euler Line, including 102 triangle centers
Algebraic Highways in Triangle Geometry
Triangle Centers and Central Triangles (book)

Unsolved Problems and Rewards
Interspersions and Dispersions
Continued Fractions
Fractal Sequences

New Harmony Scientists, Educators, Writers & Artists
Emmy Noether, Her Mentors & Colleagues
Fibonacci Number-Theorists
Triangle Geometers


Project 1. Solos for Treble Instrument, Especially Soprano Recorder
This completed project consists of 12 collections, each in two parts: Historical Notes and Solos. You can click below to access the Historical Notes, and from there, you can download most of the Solos. The others are available from Mel Bay Publications.

Collection 1: African-American and Jamaican Melodies
Collection 2: Christmas Carols (available from Mel Bay Publications.)
Collection 3: Irish Melodies
Collection 4: Americana to 1865
Collection 5: Americana after 1865
Collection 6: British Melodies
Collection 7: Melodies by Women Composers
Collection 8: Eastern European and Jewish Melodies
Collection 9: American Indian Melodies
Collection 10: Latin American Melodies
Collection 11: African Melodies
Collection 12: Western European Melodies

Project 2. Compositions for church choirs, available from G.I.A. Publications:

Four Anthems for Mixed Voices and Handbells
Scripture and Season in Song
The Hills Are Hushed This Night of Nights
O God, Beneath Your Hand
O God, Who at the Dawn of Time
Ring Out the Glad Tidings
The King of Love My Shepherd Is
This Easter Morn
Awake with Timbrel and with Dance
Introits and Anthems for Voices and Bells, Volume 1
Introits and Anthems for Voices and Bells, Volume 2
Introits and Anthems for Voices and Bells, Volume 3
Introits and Anthems for Voices and Bells, Volume 4
Introits and Anthems for Voices and Bells, Volume 5
Introits and Anthems for Voices and Bells, Volume 6

Project 3. New hymn tunes, with texts by James Hart Brumm, John Core, Ralph Freeman, Daniel P. Merrick, and John Thornburg. The book is available from Wayne Leupold Editions; click Information.

Project 4. Henry Hart, a composer who lived in Evansville, Indiana (where I live) in the 1870's. Hart published minstrel tunes and music for social dance. The research led to the first of my biographical articles for Wikipedia: Henry Hart. The External Links section at the end will take you to three supplements:

Supplement I. Henry Hart and His Family Orchestra (Including Eli Lilly's Recollections)
Supplement II. Hazel Hart Hendricks and Her School
Supplement III. Myrtle Hart, Harpist

Project 5. Articles in The Hymn, the journal of The Hymn Society in the United States and Canada:

Tunes Lost and Found: BIRSTAL, FOUNDRY, HOTHAM, TRUE ELIJAH, The Hymn 52, no. 1 (Jan., 2001) 21-30.

Tunes Lost and Found: HELMSLEY, LEEDS, YORKSHIRE, WEDNESBURY, The Hymn 52, no. 2 (Apr., 2001) 25-36.

Tunes Lost and Found: TRUMPET, LOVE-FEAST, ATHLONE, DERBY, The Hymn, no. 2 (Apr., 2002) 7-18.

The Hymn Tunes of Richard Wayne Dirksen, The Hymn 53, no. 4 (Oct., 2002) 19-28.

Hymn Tune Descants, Part 1: 1915-1934, The Hymn 54, no. 3 (July, 2003) 20-27. Reprinted in Journal of the Ralph Vaughan Williams Society 29 (Feb., 2004) 17-20.

Hymn Tune Descants, Part 2: 1935-2001, The Hymn 55, no. 1 (Jan., 2004) 17-22.

Three Native American Hymns, The Hymn 56, no. 2 (Spring 2005) 18-29. Regarding these hymns, see the Wikipedia article Joseph Renville.

Native Dakota Hymn Tunes, The Hymn 64, no. 3 (Summer 2013) 7-18.

Three Generations of Works and Their Contributions to Congregational Singing, The Hymn 65, no. 3 (Summer 2014) 10-17.

Project 6. Christmas carols for flute and accompanying instrument: Tidings of Comfort and Joy, for flute and keyboard, and 15 Christmas Carols for Flute and Guitar.

Project 7. Duets for Flutes or Recorders for flutes, recorders (bass, tenor, alto, soprano, sopranino), or other instruments: DUETS - Publisher's description.

The duets are performed on various pairs of recorders, from contrabass up to sopranino, by Papalin:
Part 1: Duets 1-27
Part 2: Duets 28-50
Part 3: Duets 51-75

Project 8. Trios for Recorders or Flutes. For downloadable free scores, click Online Sheet Music: Recorder Trios.

The 117 trios in this collection are performed on various recorders, from contrabass up to sopranino, by Papalin:
Part 1: Trios 1-27
Part 2: Trios 28-58
Part 3: Trios 59-90
Part 4: Trios 91-117

For access to the vast collection of Papalin's recordings, IL DIVO "Papalin"

Project 9. Quartets for Recorders. Scored for various ensembles of sopranino, soprano, alto, tenor, and bass recorders. For downloadable free scores, click Online Sheet Music: Recorder Quartets.

Project 10. Ten hymn tunes in In Melody and Songs, Hymns from the Psalm Versions of Isaac Watts, Adrienne Tindall, editor, Darcery Press, 2014.

Project 11. Solos for Soprano Recorder (305 advanced solos), downladable from IMSLP

Project 12. 400 Solos for C Instrument, downloadable from IMSLP

Project 13. Piano solos and a few pieces for church choirs, downloadable from IMSLP: .


The Canterbury Dictionary of Hymnology was launched from the University of Bristol in 2013. See Home Page and Editors' Introduction. My interest in hymn-tune descants, found in 1915 in England with not-yet-found French precedents, led to an article in the Journal of the Ralph Vaughan Williams Society, and from there to the Descant article in the CDH. Subsequently, I was invited to write about American hymn tune composers and early American hymn tune collections, such as William Walker's Southern Harmony. Following is a list of the resulting articles (partially downloadable to non-CDH-subscribers):

American Bass Viol
Andrew Adgate, 1762-1793
George Nelson Allen, 1812-1877
Ronald Kent Arnatt, 1930-2018
Samuel Babcock, 1760-1813
Theodore Baker, 1851-1934
Nicholas George Julius Ballanta, 1893-1961(?)
Daniel Bayley, 1729-1792
R. John Blackley, 1936-
Robert Boyd, 1771-1822?
Carrie Breck, 1855-1934
Lee Hastings Bristol, Jr., 1923-1979
Oliver Brownson, 1746-1815
Allen Carden, 1792-1859
Amzi Chapin, 1768-1835
Edwin Chapin, 1814-1880
Lucius Chapin, 1760-1842
Christian Harmony, 1867 -
Norman Coke-Jephcott, 1893-1962 -
The Colored Sacred Harp, 1934-
Harry Coopersmith, 1902-1975
Henry Stephen Cutler, 1825-1902
Daniel ben Judah, fourteenth century
Elkanah Kelsay Dare, 1782-1826
Ananias Davisson, 1780-1857
Lizzie DeArmond, 1847-1936
Descant, 1915-
Richard Dirksen, 1921-2003
Josiah Flagg, 1737-1794
Stephen Foster, 1826-1864
Philip Frazier, 1892-1964
Lewis Freeman, 1780-1859
Jacob French, 1754-1817
Harold William Friedell, 1905-1958
Clement Gale, 1860-1934
Louis Moreau Gottschalk, 1829-1869
Nathaniel Duren Gould, 1781-1864
Jean M. T. Gower, 1868-1957
John Henry Gower, 1855-1922
James Martin Gray, 1851-1935
Henry Wellington Greatorex, 1813-1858
Gerre Hancock, 1934-2012
Harp (as a title)
Karl P. Harrington, 1861-1953
William. C. Hauser, 1812-1880
Wilbur Caldwell Held, 1914-2015
Edward Hodges, 1796-1867
John Sebastian Bach Hodges, 1830-1895
Henry Houseley, 1851-1925
Matilda Durham Hoy, 1815-1901
Richard Hulan, 1939 -
Charles Lewis Hutchins, 1838-1920
Hymntune Index and Related Hymn Materials
Jeremiah Ingalls, 1764-1838
Simeon Jocelin, 1746-1823
Artemas N. Johnson, 1817-1892
Julia H. Johnston, 1849-1919
Kentucky Harmony and Supplement, 1816 -
Elisha J. King, 1821?-1844
Karl Douglas Kroeger, 1932 -
Chaunc(e)y Langdon, 1763-1830
Andrew Law, 1749-1821
James Lyon, 1735-1794
Albert Hay Malotte, 1895, 1964
Malotte's Lord's Prayer, 1935
Elias Mann, 1750-1825
Many and great, O God, are thy works, 1846
Cotton Mather, 1663-1728
Abraham Maxim, 1773-1829
Jesse Mercer, 1769-1841
Mercer's 'Cluster'
Arthur H. Messiter, 1834-1916
Missouri Harmony, 1820 -
Mary Whittle Moody, 1870-1963
James R. Murray, 1841-1905
John Murray, 1840-1915
Judith Murray, 1751-1820
John Jacob Niles, 1892-1980
Charles S. Nutter, 1842-1928
Philip Phillips, 1834-1895
Harriet H. Pierson, 1865-1921
Amos Pilsbury, 1772-1812
George Frederick Root, 1812-1895
The Sacred Harp, 1844 -
Alexander Schreiner, 1901-1987
Shenandoah Harmony, 2012
Mary Shindler, 1810-1883
Nehemiah Shumway, 1761-1843
Southern Harmony, 1835 -
Harriet Spaeth, 1845-1925
John Stickney, 1744-1827
Timothy Swan, 1758-1843
Elizabeth Miller Tangeman, 1905-1996
Reubern Archer Torrey, 1856-1928
Daniel B. Towner, 1850-1919
David Elton Trueblood, 1900-1994
James Ireland Tucker, 1819-1895
William Tuckey, 1708-1781
John Tufts, 1689-1750
Grant Colfax Tullar, 1869-1950
Samuel Wakefield, 1799-1895
William Walker, 1809-1875
Thomas Walter, 1696-1725
William Henry Walter, 1825-1893
Samuel Augustus Ward, 1847-1903
Henry Ware, Jr., 1794-1843
Donald DeWitt Wasson, 1921-2009
George James Webb, 1803-1887
George Whelpton, 1847-1930
Benjamin Franklin White, 1800-1879
David McK. Williams, 1887-1978
Frederick Jerome Work, 1878?-1942
John Wesley Work (I), 1848?-1923
John Wesley Work (II), 1872?-1923
John Wesley Work (III), 1901-1967
Wyeth's Repository of Sacred Music, 1813


Neil Sloane's Online Encyclopedia of Integer Sequences (OEIS) is a treasury of mathematical patterns, including references, links, and comments. Many contributors have developed prototypical sequences that introduce newly discovered relationships among numbers, some proved and others conjectured. Here are several:

Accumulation arrays and chains, weight arrays: A144112, A185906, A185778
Adjective-noun sequences and conjectures: A055168- A055185 and A055186- A055191
Adjusted joint rank sequences: A186219, A186221, A186350
Algorithm ("rules 1,2,3", conjectured permutations of N): A257705,
A257905, A258046
Arrays defined by recurrences: A026519, A026568, A027926, A246694 (every positive integer exactly twice), A246696 (every positive integer exactly once)
Augmentation arrays: A193091
Averaging arrays: A197737
Beatty arrays: A181661, A181886, A183988
Beatty discrepancy, complementary equations: A136496, A138253, A138330
Beatty trees and back-stepping: A258212
Best remaining lower approximates to golden ratio: A143514
Change-of-base functions (geometric fractals): A214969
Coefficient arrays from recursively defined polynomials: A249057, A249100, A249128, A249250
Complementary equations: A022424, A293076, A293358, A294381, A294414, A294532, A295053, A295357, A295862, A296000, A296245, A297830, A298000, A299634, A304799 (advanced subscripts), A335999
Constant associated with fractal change-of-base functions: A188593,
Constants associated with special shapes: A188593, A188618, A188635, A195284 (Philo distance, incenter), A195304 (Philo distance, centroid), A248965, A248967, A248968, A336043, A336073,
Constants associated with extreme values: A196500
Constant associated with continued fraction and Hofstadter eta sequence: A245215
Constants associated with convergent sums of reciprocals: A228040, A228044
Constants associated with the Lambert W funciton (PowerLog): A299613, A228044
Constants associated with tangencies: A196625, A197016
Constants that solve transcendental equations: A196396, A196401, A196407, A197737, A197739
Constants defined by limits: A298512
Continued fractions, accelerated: A228667
Continued fractions, delayed: A228825
Continued fractions (including conjectures):
     A013659 (initialized continued fractions)
     A096654- A096658 (self-generating continued fractions)
     A143515 (best remaining upper approximates)
     A143527 (best lower remaining approximates)
     A229287 (primes)
     A229348 (Fibonacci-ratio terms)
     A229351 (fractional terms)
     A229981 (associated with divergent continued fraction)
     A229983 (associated with divergent continued fraction)
     A229985 (associated with divergent continued fraction)
     A229987 (associated with divergent continued fraction)
     A230000 (continued fraction [1,1/x,1/x^2,...])
     A259588 ("other-side convergents")
     A259911 (discriminants of purely periodic continued fractions)
     A265288 (sum{x - c(2n-1), n=1,2,...}, where c = convergents to (x = golden ratio))
     A265762 (coefficients in minimal polynomial of [1^n,2,1,1,1...])
Continued fraction transform: A229594
Convolution arrays: A213500
Convolutory inverses: A077607, A078140, A225127
Convolutory roots: A255811
Digits in various bases (pits and peaks, rises and falls, etc.)_ A296712 A296882, A297030, A297146, A297330
Dispersions: A114537 (the primes), A191426, A248513
Divided difference sequences: A204892
Double interspersion: A161179
Egyptian fractions (denominates, convergent series): A226049
Euclidean algorithm, modified, accelerated (see also Continued fractions, accelerated, delayed): A228247, A228468 (definition of trace of (x,y)), A228486 (definition of trace of a number, in particular, pi), A228487, A228488, A269993
Iteration from natural numbers to lower Wythoff sequence: A181108
Factorial-nested interval sequences: A269970, A269982(definition of factorial fractility)
Fibonacci-related iterations: A252230
Fibonacci regression array: A097367
Fibonacci (minimal alternating representations of nonnegative integers, using Fibonacci numbers): A256655
Fission arrays (of polynomial sequences, infinite matrices) A193842, A193844 (Pascal self-fission array), A202503 (Fibonacci self-fission array)
Floor-algorithm (steps from n to 1): - A257698-A257704
Floor-sum sequence: A182653
Floor-related recurrence sequences: A298338
Fractal sequences: A003602, A194029, A194832, A248514 A272900
Fractalization: A194959
Fractional parts (counts involving inequalities, intervals): A194285, A194738, A194813
Fractional parts (difference between): A194368
Fusion arrays (of polynomial sequences, infinite matrices) A118800 (Pascal self-fusion array), A193722, A202453 (Fibonacci self-fusion array)
Golden ratio: A252229,
Golden triangle: A152149, A178988, A188635
Greatest cube that divides n!: A248762
Greedy sequences: A270744
Greedy residue sequences: A241833 (greedy residue sequence of squares), A241834 (of triangular numbers), A242252 (of odd primes), A242293 (of cubes)
Harmonic mean: A227012 (includes conjectures)
Harmonic numbers:
     A088907 (least k such that mean(1,1/2,...,1/k)<=1/n)
     A224820 (array, includes open question about linear recurrence)
     A225918 (partial sums of harmonic series),
     A226894 (joint ranking of {H(k)} and {log k})
     A227581 (closeness of H(m*n) to H(m) + H(n))
     A227631 (definition of least splitter, a conjecture)
Infinite Fibonacci word (related sequences): A241422, A245920, A245975, A246104, A246127, A246354
Inflection point of product{1 + x^k, k >=1}: A257394
Joint-rank arrays: A184912- A184927, A182801, A182869
Jointly generated arrays (by paired polynomial recurrences): A208510
Jointly ranked sequences (with a conjecture): A206911
Jump sequences (involving floor and ceiling, with conjectures): A219085
Kimberling expulsion array: A006852, A007963, A035486
Knight's moves on infinite chessboard: A183049
Kolakoski fan conjectures: A144586, A143589
Lagrange pairs: A194508
Lattice points in a region, counted: A227347
Lazy Fibonacci representations: A095791, A095903
Least k such that e - (1 + 1/k)^k < 1/n: A247778
Least k such that 1/e - (1 - 1/k)^k < 1/n: A247781
Least k such that e - 1/((k!)^(1/k)) < 1/n: A247988
Least k such that k*sin(1/k)-1 < 1/n^2: A248515
Least number of (or not of) a prescribed form: A305328, A325417
Limit involving complex numbers and continued fractions: A248749
Limit-reverse of words:
     A245920 (of infinite Fibonacci word)
     A245933 (of Beatty sequence for sqrt(2))
     A245936 (of Oldenburger-Kolakoski sequence)
     A245939 (of Thue-Morse sequence)
Limiting block extension: A246127
Limiting counting procedure (Theorems 1 and 2): A225660, A225669
Matrix counts: A194508
Minimal alternating representations of nonnegative integers, using various bases: A256655 (Fibonacci numbers), A256696 (powers of 2), A256974 (triangular numbers), A256789 (square numbers),
Morphism sequences: A189576, A189628, A287663, A287385
Multinomial coefficients: A248686
Mysterious B-sequence (array): A186158
Natural number array: A000027, A185787
Number of diophantine solutions (w,x,y,n), etc.: A211266, A211422, A211795, A212959
Order arrays: A163254, A163255
Ordering of rational numbers and other sets; see also "Trees generated by various sequences" and "Rules-generated sequences" :
     A226080 (Fibonacci - or rabbit - ordering, all positive integers)
     A226130 (all rationals)
     A226247 (all rationals)
     A232559 (all positive rationals)
     A232723 (all integers)
     A274455 (all integers)
     A232895 (all positive integers; linear recurrence conjecture)
     A232890 (all rationals)
     A233694 (all Gaussian rationals)
P-INVERT of a sequence: A289780, A290890 A291000, A291219, A291382, A291728, A292479 A292480
Palindromic linear recurrences; array R(n,k) = n+[n/2]+...[n/k]: A211701
Palindromic words (infinite): A260390
Partial complements, polygonal numbers: A086270, A086272
Partitions of the set of positive integers ("generalized Beatty sequences", joint rankings): A184812, A184928
Partitions of positive integers:      A184812
     A209082 (least power separator)
     A209122 (# partitions into k parts is k for some k)
     A209354 (max - min)
     A209423 (# odd parts minus # even parts)
     A209489 (partitions of 1 into rational parts)
     A230025 (outliers in partitions)
     A235130 (partitions that include partitions)
     A236129 (constant reversal sums)
     A236292 (distinct cyclic permutations of partitions of n)
     A236360 (numerator, mean of all parts of partitions of n)
     A236543 (least part and greatest part, frequencies)
     A237665 (string of consecutive integers)
     A237668 (sums of parts is a part)
     A237685 (depth of a partition)
     A237980 (square partitions)
     A237981 (directional partitions; definition of Ferrers matrix)
     A238325 (antidiagonal partitions)
     A238326 (diagonal partitions)
     A238616 (partitions with standard deviation < 1)
     A238883 (upper triangular partitions)
     A238885 (lower triangular partitions)
     A238943 (size of Ferrers matrix of partitions)
     A239312 (condensed partitions)
     A239319 (condensed partitions)
     A239482 (separable partitions)
     A241150 (partition graph, G(n))
     A241815 (strict partition graph, S(n))
     A241900 (components of disconnected partition graph, S(n))
Partitions of zero: A209535, A236429
Path-counting arrays: A247049, A247352
Permutations of integers (conjectures): A131388- A131395
Polynomial reduction: A192175, A192232, A192744, A192872, A192904, A192951
Polynomials defined by recurrences: A184049, A240755, A240758, A248664, A248669, A248670
Power ceiling functions: A214992 (several open questions)
Primes (conjectures): A174349, A184774, A185510, A219096, A258036, A255909
Prime divisor subtractions iterated until reaching 0: A233341
Prime-related arrays: A192175, A229608
Primes, a limit: A260458 (limit of GCD(PP(n) - k, PP(n) + k) as k -> oo, where PP(n) is the product of the first n primes.
Primes, minimal nested palindromic primes: A261881, A262627, A260444 (infinite palindromic word)
Primes in various bases: A262727 (2,3,5)-primes, A262728 (2,3,5,7)-primes, A262729 strong (2,3,5,7)-primes
Primes-only best approximates (POBAs) to 1: A265759
Pythagorean fraction approximations: A195500
k-Pythagorean triples: A195770
Q-residues of polynomials: A193649
Quarter-squares, greedy representations of positive integers (with theorem): A257018
Rank transforms: A187224
Ratio-sums: A296469
Reciprocity arrays: - A259572-A259583
Repetition-resistant sequence: A007061, A079101
Rules-generated sequences: A074049, A147991, A183544, A190803, A191106
Self-generating arrays: A203300, A203301
Shift-sums of zero-one sequences: A187950, A188014
Signature sequences: A007337, A084531
Subsets counted by maximal differences: A255874
Swappage (lower even swappage of upper Wythoff sequence): A141104
Symmetric matrices and interlacing roots of polynomials: A202605, A204016
Symmetric matrices and permanents: A204263, A204269, A204435, A204551
Trees generated by rules about edges: A228853
Trees generated by various sequences or rules:
     A074049 (generated by Wythoff sequences)
     A183079 (generated by triangular numbers)
     A183169 (generated by squares)
     A183231 (1st of two complementary trees generated by triangular numbers)
     A183542 (1st of two complementary trees generated by Wythoff sequences
     A233135 (shortest (x+1,2x)-code of n)
     A252864 (a tree of ordered pairs)
     A255809 (generation n has n-tuples with gapsizes <=2)
     A257569 (Fibonacci-related tree)
     A262841 (Fibonacci zero-tree)
     A264799 (every integer exactly once)
Unimodular 2x2 matrices: A210000
Vandermonde permanents: A203468
Wythoff array, W: A035513, A114579 (transposition sequence of W), A185778, A186007, A259556
Wythoff difference array: A080164
s-Wythoff sequences: A184117
Zeckendorf distance between m and n: A226207
Zero-one sequences (self-generating): A188967



A pdf of The Shape and History of The Ellipse in Washington, D. C. (Click!)