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From M. F. Hasler, Feb 16 2023: (Start)

The numbers that appear 3 times in Pascal's triangle are the central binomial coefficients (A000984), except for the number 2 that is the only number to appear only once. For n = 1 there would be an infinite number of occurrences, but sequence A003016 counts only the occurrences of n in rows <= n so that n = 1 also gives 3.

All C(n,k) with 1 < k < n/2 (in particular triangular numbers A000217) appear at least 4 times; see A098564 for those appearing exactly 4 times.

Numbers that appear 5 or more times are quite rare, they are listed in A003015 with subsequence A098565 of those appearing exactly 6 times.

They are mostly C(n,k) with 2 < k < n/2 which are also triangular numbers, but some are also of the form C(n+1,k) = C(n,k+1) with 3 < k < n/2, and a subsequence of these has n and k given in terms of Fibonacci numbers. (End)

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

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

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

See the b-file of A090162 for the explicit numbers produced by the parametric formula. - Jeppe Stig Nielsen, Aug 23 2020

From the following 4 disjoint subsets of natural numbers A = {1}, B = {2}, OP = {odd primes}, C = {composites}, 16 sets are derivable: A000027 versus empty set, A002808 vs A008578, A065091 vs A065090, A000040 vs A018252, A006005 vs {{2} with A002808}, {1} vs {A000027 excluding 1}, {2} versus this sequence, {1, 2} versus Union[OP, C].

a(n) is the sum of the obvious divisors of n, which are 1 and n.

The natural numbers excluding 2 are the order numbers of magic squares. Order 2 magic squares do not exist. - William Walkington, Mar 12 2016

The numbers occurring at least twice in Pascal's triangle (A007318, A003016). - Rick L. Shepherd, Jun 05 2016

From Enrique Navarrete, Mar 03 2025: (Start)

a(n) is the number of binary strings of length n with at most one 0 and at least one 1. For example, the a(1)=1 string is 1 and the a(2)=3 strings are 01, 10, 11.

a(n) is also the number of ordered set partitions of an n-set into 2 sets such that the first set has at most one element and the second set has at least one element. (End)

Central binomial coefficients c = A000984(n) > 1 appear once in the middle column C(2n, n), and thereafter in one or more later rows to the left as C(r,k) and to the right as C(r, r-k), k < r/2; the last time in row r = c = C(c,1) = C(c,c-1). For these, a(n) = (A003016(n)+1)/2. For all other numbers n > 1, a(n) = A003016(n)/2. - M. F. Hasler, Mar 01 2023

A059233(a(n)) = n and A059233(m) < n for m < a(n).

From the formula, if we know k, we can estimate n as approximately 0.83 sqrt(k).

Open question: Does binomial(2*a(n) + 1, a(n) - n) = binomial(2*a(n), a(n)) for any n? An affirmative answer would settle whether there exists an odd term greater than 3 in A003016. - Danny Rorabaugh, Mar 16 2016

Let # denote "number of" and c(p) = conjugate of partitionp. Then

A240057(n) = # p such that min(p) not = max(c(p));

A039899(n) = # p such that min(p) < max(c(p));

A039900(n) = # p such that min(p) <= max(c(p));

A006141(n) = # p such that min(p) = max(c(p));

A003114(n) = # p such that min(p) > max(c(p));

A003016(n) = # p such that min(p) >= max(c(p));

A064173(n) = # p such that max(p) < max(c(p));

A064174(n) = # p such that max(p) <= max(c(p));

A047993(n) = # p such that max(p) = max(c(p)).

See A240178 for related sequences. - Clark Kimberling, Apr 11 2014

a(n) >= A001227(n) + A307666(n).

a(n) >= A003016(n) + A003016(n+1) - 2.

Records occur at indices 2, 3, 5, 6, 9, 10, 15, 35, 55, 105, 210, 1365, 2925, 3003,...

In other words, the n-th rows lists the numbers that appear directly above n in Pascal's triangle (A007318).

The n-th row starts with 1, ends with n-1 (provided that n > 1), and contains other values iff n belongs to A006987.