1,2

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 {n choose m-1} = {n-1 choose 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. - Chris Thompson (cet1(AT)cam.ac.uk), 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 (njas(AT)research.att.com), Nov 24 2004

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

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

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

B. M. M. de Weger, Equal binomial coefficients: some elementary considerations (Postscript)

Cf. A003016, A059233.

Sequence in context: A069790 A064224 A069674 this_sequence A098565 A084142 A146950

Adjacent sequences: A003012 A003013 A003014 this_sequence A003016 A003017 A003018

nonn

N. J. A. Sloane (njas(AT)research.att.com).

No other terms below 33*10^16 (David W. Wilson (davidwwilson(AT)comcast.net)).

61218182743304701891431482520 really is the next term. Weger shows this and I checked it. - T. D. Noe (noe(AT)sspectra.com), Nov 15 2004