editing

proposed

W. William A. Stein, <a href="httphttps://wstein.org/edu/Fall2001/124/lectures/lecture6/html/node3.html">Phi is a Multiplicative Function</a>

A. de Vries, <a href="http://math-it.org/Mathematik/Zahlentheorie/Zahl/ZahlApplet.html">The prime factors of an integer (along with Euler's phi and Carmichael's lambda functions)</a>

G. Gang Xiao, Numerical Calculator, <a href="httphttps://wims.uniceuniv-cotedazur.fr/wims/en_tool~number~calcnum.en.html">To display phi(n) operate on "eulerphi(n)"</a>.

D. Derrick N. Lehmer, <a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-26/issue-3/Dicksons-History-of-the-Theory-of-Numbers/bams/1183425137.full">Review of Dickson's History of the Theory of Numbers</a>, Bull. Amer. Math. Soc., 26 (1919), 125-132.

K. Keith Matthews, <a href="http://www.numbertheory.org/php/factor.html">Factorizing n and calculating phi(n), omega(n), d(n), sigma(n) and mu(n)</a>.

WWacław F. Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4206.pdf">Euler's Totient Function And The Theorem Of Euler</a>.

N. Neil J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence)

N. Neil J. A. Sloane, <a href="https://arxiv.org/abs/2301.03149">"A Handbook of Integer Sequences" Fifty Years Later</a>, arXiv:2301.03149 [math.NT], 2023, p. 14.

U. Ulrich Sondermann, <a href="https://web.archive.org/web/20110823215228/http://home.earthlink.net/~usondermann/eulertot.html">Euler's Totient Function</a>.

M. Milton Abramowitz and I. Irene A. Stegun, eds., <a href="https://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972.

DDario A. Alpern, <a href="https://www.alpertron.com.ar/ECM.HTM">Factorization using the Elliptic Curve Method (along with sigma_0, sigma_1 and phi functions)</a>

A. Alexander Bogomolny, <a href="httphttps://www.cut-the-knot.org/blue/Euler.shtml">Euler Function and Theorem</a>.

C. Chris K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php?sort=EulersPhi">Euler's phi function</a>

R. Robert D. Carmichael, <a href="/A002180/a002180.pdf">A table of the values of m corresponding to given values of phi(m)</a>, Amer. J. Math., 30 (1908), 394-400. [Annotated scanned copy]

K. Kevin Ford, <a href="https://arxiv.org/abs/math/9907204">The number of solutions of phi(x)=m</a>, arXiv:math/9907204 [math.NT], 1999.

E. Pérez Herrero, <a href="httphttps://psychedelic-geometry.blogspot.com/2010/07/totient-carnival.html">Totient Carnival partitions</a>, Psychedelic Geometry Blogspot.

M. Lal and P. Gillard, <a href="httphttps://dx.doi.org/10.1090/S0025-5718-69-99858-5">Table of Euler's phi function, n < 10^5</a>, Math. Comp., 23 (1969), 682-683.

approved

editing

editing

proposed

P. Phil Lafer, <a href="httphttps://www.fq.math.ca/Scanned/9-1/lafer.pdf">Discovering the square-triangular numbers</a>, Fib. Quart., 9 (1971), 93-105.

Kalman Liptai, <a href="httphttps://www.fq.math.ca/Papers1/42-4/quartliptai04_2004.pdf">Fibonacci Balancing Numbers</a>, Fib. Quart. 42 (4) (2004) 330-340.

Madras College, St Andrews, <a href="https://web.archive.org/web/20190920231615/http://www.madras.fife.sch.uk:80/departments/Mathematics/activities/amazingnofacts/fact017.html">Square Triangular Numbers</a>

Roger B. Nelson, <a href="httphttps://www.jstor.org/stable/10.4169/math.mag.89.3.159">Multi-Polygonal Numbers</a>, Mathematics Magazine, Vol. 89, No. 3 (June 2016), pp. 159-164.

G. K. Panda and S. S. Rout, <a href="httphttps://dx.doi.org/10.1007/s10474-014-0427-z">Periodicity of Balancing Numbers</a>, Acta Mathematica Hungarica 143 (2014), 274-286.

Poo-Sung Park, <a href="httphttps://www.jstor.org/stable/30044886">Ramanujan's Continued Fraction for a Puzzle</a>, College Mathematics Journal, 2005, 363-365.

A. Sandhya, <a href="httphttps://www.angelfire.com/ak/ashoksandhya/maths2.html">Puzzle 4: A problem Srinivasa Ramanujan, the famous 20th century Indian Mathematician Solved</a>

A. Ahmet Tekcan, M. Merve Tayat, and M. Meltem E. Ozbek, <a href="httphttps://dx.doi.org/10.1155/2014/897834">The diophantine equation 8x^2-y^2+8x(1+t)+(2t+1)^2=0 and t-balancing numbers</a>, ISRN Combinatorics, Volume 2014, Article ID 897834, 5 pages.

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/BinomialCoefficient.html">Binomial coefficient</a>.

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/SquareTriangularNumber.html">Square Triangular Number</a>.

Eric Weisstein's World of Mathematics, <a href="httphttps://mathworld.wolfram.com/TriangularNumber.html">Triangular Number</a>.

Wikipedia, <a href="httphttps://en.wikipedia.org/wiki/Triangular_square_number">Triangular square number</a>

Aviezri S. Fraenkel, <a href="httphttps://dx.doi.org/10.1016/S0012-365X(00)00138-2">On the recurrence f(m+1)= b(m)*f(m)-f(m-1) and applications</a>, Discrete Mathematics 224 (2000), pp. 273-279.

Robert Frontczak, <a href="httphttps://www.m-hikari.com/ijma/ijma-2018/ijma-9-12-2018/p/frontczakIJMA9-12-2018.pdf">Sums of Balancing and Lucas-Balancing Numbers with Binomial Coefficients</a>, International Journal of Mathematical Analysis (2018) Vol. 12, No. 12, 585-594.

Bill Gosper, <a href="httphttps://gosper.org/triangsq.pdf">The Triangular Squares</a>, 2014.

H. Harborth, <a href="httphttps://dx.doi.org/10.1007/978-94-015-7801-1_1">Fermat-like binomial equations</a>, Applications of Fibonacci numbers, Proc. 2nd Int. Conf., San Jose/Ca., August 1986, 1-5 (1988).

D. B. Brian Hayes, <a href="httphttps://www.americanscientist.org/libraries/documents/200884115366940-2008-09Hayes.pdf">Calculemus!</a>, American Scientist, 96 (Sep-Oct 2008), 362-366.

M. Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Janjic/janjic63.html">On Linear Recurrence Equations Arising from Compositions of Positive Integers</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.

Michael A. Jones, <a href="httphttps://www.jstor.org/stable/10.4169/college.math.j.43.3.212">Proof Without Words: The Square of a Balancing Number Is a Triangular Number</a>, The College Mathematics Journal, Vol. 43, No. 3 (May 2012), p. 212.

O. Omar Khadir, K. Kalman Liptai, and L. Laszlo Szalay, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Szalay/szalay11.html">On the Shifted Product of Binary Recurrences</a>, J. Int. Seq. 13 (2010), 10.6.1.

Tanya Khovanova, <a href="httphttps://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>

P. Paula Catarino, H. Helena Campos, and P. Paulo Vasco, <a href="http://ami.ektf.hu/uploads/papers/finalpdf/AMI_45_from11to24.pdf">On some identities for balancing and cobalancing numbers</a>, Annales Mathematicae et Informaticae, 45 (2015) pp. 11-24.

Tomislav Doslic, <a href="httphttps://dx.doi.org/10.1007/s10910-013-0167-2">Planar polycyclic graphs and their Tutte polynomials</a>, Journal of Mathematical Chemistry, Volume 51, Issue 6, 2013, pp. 1599-1607.

D. B. Eperson, <a href="httphttps://www.jstor.org/stable/3613402">Triangular numbers</a>, Math. Gaz., 47 (1963), 236-237.

L. Leonhard Euler, <a href="httphttps://mathscholarlycommons.dartmouthpacific.edu/~euler-works/pages29/E029.html">De solutione problematum diophanteorum per numeros integros</a>, Par. 19.

S. Sergio Falcon, <a href="httphttps://dx.doi.org/10.4236/am.2014.515216">Relationships between Some k-Fibonacci Sequences</a>, Applied Mathematics, 2014, 5, 2226-2234.

R. Rigoberto Flórez, R. Robinson A. Higuita, and A. Antara Mukherjee, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mukherjee/mukh2.html">Alternating Sums in the Hosoya Polynomial Triangle</a>, Article 14.9.5 Journal of Integer Sequences, Vol. 17 (2014).

I. Irving Adler, <a href="http://www.fq.math.ca/Scanned/7-2/adler.pdf">Three Diophantine equations - Part II</a>, Fib. Quart., 7 (1969), pp. 181-193.

Dario Alpern for Diophantine equation <a href="httphttps://www.alpertron.com.ar/SUMPOWER.HTM#4_3_2">a^4+b^3=c^2</a>.

K. Kasper Andersen, L. Lisa Carbone, and D. Penta, <a href="https://pdfs.semanticscholar.org/8f0c/c3e68d388185129a56ed73b5d21224659300.pdf">Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields</a>, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.

A. Behera and G. K. Panda, <a href="httphttps://www.fq.math.ca/Scanned/37-2/behera.pdf">On the Square Roots of Triangular Numbers</a>, Fib. Quart., 37 (1999), pp. 98-105.

Elwyn Berlekamp and Joe P. Buhler, <a href="httphttps://www.msri.org/attachments/media/news/emissary/EmissaryFall2005.pdf">Puzzle Column</a>, Emissary, MSRI Newsletter, Fall 2005. Problem 1, (6 MB).

D. Daniel Birmajer, J. Juan B. Gil, and M. Michael D. Weiner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Gil/gil6.html">On the Enumeration of Restricted Words over a Finite Alphabet</a>, J. Int. Seq. 19 (2016) # 16.1.3, example 12.

A. Alexander Bogomolny, <a href="httphttps://www.cut-the-knot.org/do_you_know/triSquare.shtml">There exist triangular numbers that are also squares</a>

John C. Butcher, <a href="httphttps://www.math.auckland.ac.nz/~butcher/miniature/miniature2.html">On Ramanujan, continued Fractions and an interesting number</a>

approved

editing

editing

proposed