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A003015 Numbers that occur 5 or more times in Pascal's triangle.
(Formerly M5374)
1, 120, 210, 1540, 3003, 7140, 11628, 24310, 61218182743304701891431482520 (list; graph; refs; listen; history; text; internal format)



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. - 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


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).


Table of n, a(n) for n=1..9.

Aart Blokhuis, Andries Brouwer, Benne de Weger, Binomial collisions and near collisions, arXiv:1707.06893 [math.NT], 2017.

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


Cf. A003016, A059233.

Cf. A182237, A098565 (subsequence).

Sequence in context: A069790 A064224 A069674 * A098565 A084142 A256814

Adjacent sequences:  A003012 A003013 A003014 * A003016 A003017 A003018




N. J. A. Sloane



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Last modified August 23 19:47 EDT 2017. Contains 291021 sequences.