A003015 Numbers that occur 5 or more times in Pascal's triangle.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

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

No other terms below 33*10^16 (David W. Wilson).

61218182743304701891431482520 really is the next term. Weger shows this and I checked it. - T. D. Noe, Nov 15 2004

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

See the b-file of A090162 for the explicit numbers produced by the parametric formula. - Jeppe Stig Nielsen, Aug 23 2020

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 93, #47.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Aart Blokhuis, Andries Brouwer, and Benne de Weger, Binomial collisions and near collisions, INTEGERS, Volume 17, Article A64, 2017 (also available as arXiv:1707.06893 [math.NT]).
• Jean-Marie de Koninck, Nicolas Doyon, and William Verreault, Repetitions of multinomial coefficients and a generalization of Singmaster's conjecture, Integers (2021) Vol. 21, #A34.
• B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Econometric Institute Research Papers, No. EI 9536-/B, 1995.
• B. M. M. de Weger, Equal binomial coefficients: some elementary considerations, Journal of Number Theory, Volume 63, Issue 2, April 1997, Pages 373-386.
• Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video (2020).
• R. K. Guy and V. Klee, Monthly research problems, 1969-1971, Amer. Math. Monthly, 78 (1971), 1113-1122.
• D. A. Lind, The quadratic field Q(sqrt(5)) and a certain diophantine equation, Fibonacci Quart. 6 (3) (1968), 86-93.
• Kaisa Matomäki, Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen, Singmaster's conjecture in the interior of Pascal's triangle, arXiv:2106.03335 [math.NT], 2021.
• Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 69.
• David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298.
• Eric Weisstein's World of Mathematics, Pascal's Triangle
• Tomohiro Yamada, Necessary conditions for binomial collisions, arXiv:2002.07043 [math.NT], 2020.

Crossrefs

Cf. A003016, A059233.

Cf. A182237, A098565 (subsequence).

Cf. A090162 (easy subsequence).