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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 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, Benne de Weger, Binomial collisions and near collisions, INTEGERS, Volume 17, Article A64, 2017 (also available as arXiv:1707.06893 [math.NT]). 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. R. K. Guy and V. Klee, Monthly research problems, 1969-1971, Amer. Math. Monthly, 78 (1971), 1113-1122. David Singmaster, Repeated binomial coefficients and Fibonacci numbers, Fibonacci Quarterly 13 (1975) 295-298. Eric Weisstein's World of Mathematics, Pascal's Triangle CROSSREFS Cf. A003016, A059233. Cf. A182237, A098565 (subsequence). Sequence in context: A069790 A064224 A069674 * A098565 A084142 A256814 Adjacent sequences:  A003012 A003013 A003014 * A003016 A003017 A003018 KEYWORD nonn AUTHOR STATUS approved

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Last modified October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)