

A003015


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


17




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,m1) = binomial{n1,m) given by n = F_{2k}F_{2k+1}; m = F_{2k1}F_{2k} where F_i is the ith 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
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

Zoe Griffiths, My MegaFavNumber: 61,218,182,743,304,701,891,431,482,520, YouTube video (2020).


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



