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A003015 Numbers that occur 5 or more times in Pascal's triangle.
(Formerly M5374)
16

%I M5374

%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

%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, 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 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 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 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>

%Y Cf. A003016, A059233.

%Y Cf. A182237, A098565 (subsequence).

%K nonn

%O 1,2

%A _N. J. A. Sloane_

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Last modified November 14 01:24 EST 2019. Contains 329108 sequences. (Running on oeis4.)