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 A003015 Numbers that occur 5 or more times in Pascal's triangle. (Formerly M5374) 16
 1, 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 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. 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). Sequence in context: A069790 A064224 A069674 * A098565 A084142 A349745 Adjacent sequences:  A003012 A003013 A003014 * A003016 A003017 A003018 KEYWORD nonn AUTHOR STATUS approved

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Last modified September 29 01:05 EDT 2022. Contains 357082 sequences. (Running on oeis4.)