%I M5374 #83 Aug 01 2024 05:34:51

%S 1,120,210,1540,3003,7140,11628,24310,61218182743304701891431482520

%N Numbers that occur 5 or more times in Pascal's triangle.

%C The subject of a recent thread on sci.math. Apparently it has been known for many years that there are infinitely many numbers that occur at least 6 times in Pascal's triangle, namely the solutions to binomial(n,m-1) = binomial(n-1,m) given by n = F_{2k}*F_{2k+1}; m = F_{2k-1}*F_{2k} where F_i is the i-th Fibonacci number. The first of these outside the range of the existing database entry is {104 choose 39} = {103 choose 40} = 61218182743304701891431482520. - _Christopher E. Thompson_, Mar 09 2001

%C It may be that there are no terms that appear exactly 5 times in Pascal's triangle, in which case the title could be changed to "Numbers that occur 6 or more times in Pascal's triangle". - _N. J. A. Sloane_, Nov 24 2004

%C No other terms below 33*10^16 (_David W. Wilson_).

%C 61218182743304701891431482520 really is the next term. Weger shows this and I checked it. - _T. D. Noe_, Nov 15 2004

%C Blokhuis et al. show that there are no other solutions less than 10^60, nor within the first 10^6 rows of Pascal's triangle other than those given by the parametric solution mentioned above. - _Christopher E. Thompson_, Jan 19 2018

%C See the b-file of A090162 for the explicit numbers produced by the parametric formula. - _Jeppe Stig Nielsen_, Aug 23 2020

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Aart Blokhuis, Andries Brouwer, and Benne de Weger, <a href="http://math.colgate.edu/~integers/vol17.html">Binomial collisions and near collisions</a>, INTEGERS, Volume 17, Article A64, 2017 (also available as <a href="https://arxiv.org/abs/1707.06893">arXiv:1707.06893 [math.NT]</a>).

%H Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, <a href="http://math.colgate.edu/~integers/v34/v34.mail.html">Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture</a>, Integers (2021) Vol. 21, #A34.

%H B. M. M. de Weger, <a href="http://hdl.handle.net/1765/1356">Equal binomial coefficients: some elementary considerations</a>, Econometric Institute Research Papers, No. EI 9536-/B, 1995.

%H B. M. M. de Weger, <a href="http://dx.doi.org/10.1006/jnth.1997.2109">Equal binomial coefficients: some elementary considerations</a>, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 373-386.

%H Zoe Griffiths, <a href="https://www.youtube.com/watch?v=Z3xq4ODNeZs">My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520</a>, YouTube video (2020).

%H R. K. Guy and V. Klee, <a href="http://www.jstor.org/stable/2316321">Monthly research problems</a>, 1969-1971, Amer. Math. Monthly, 78 (1971), 1113-1122.

%H D. A. Lind, <a href="https://www.fq.math.ca/Scanned/6-3/6-3/lind.pdf">The quadratic field Q(sqrt(5)) and a certain diophantine equation</a>, Fibonacci Quart. 6 (3) (1968), 86-93.

%H Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, <a href="https://arxiv.org/abs/2106.03335">Singmaster's conjecture in the interior of Pascal's triangle</a>, arXiv:2106.03335 [math.NT], 2021.

%H Hans Montanus and Ron Westdijk, <a href="https://greenbluemath.nl/wp-content/uploads/2024/03/Cellular-Automation-and-Binomials.pdf">Cellular Automation and Binomials</a>, Green Blue Mathematics (2022), p. 69.

%H David Singmaster, <a href="http://www.fq.math.ca/Scanned/13-4/singmaster.pdf">Repeated binomial coefficients and Fibonacci numbers</a>, Fibonacci Quarterly 13 (1975) 295-298.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PascalsTriangle.html">Pascal's Triangle</a>

%H Tomohiro Yamada, <a href="https://arxiv.org/abs/2002.07043">Necessary conditions for binomial collisions</a>, arXiv:2002.07043 [math.NT], 2020.

%Y Cf. A003016, A059233.

%Y Cf. A182237, A098565 (subsequence).

%Y Cf. A090162 (easy subsequence).

%K nonn

%O 1,2

%A _N. J. A. Sloane_