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

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.

Adjacent sequences: A003012 A003013 A003014 this_sequence A003016 A003017 A003018

Sequence in context: A069790 A064224 A069674 this_sequence A098565 A084142 A146950

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