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%I #7 Mar 01 2023 15:34:21
%S 0,1,10,120,3003
%N Where record values occur in A003016.
%C It appears that the record values are 0, 3, 4, 6, 8, 10, 12, ...
%C From _M. F. Hasler_, Feb 16 2023: (Start)
%C 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.
%C 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.
%C Numbers that appear 5 or more times are quite rare, they are listed in A003015 with subsequence A098565 of those appearing exactly 6 times.
%C 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)
%o (PARI) m=-1; [n | n<-[0..9999], m < m = max(A003016(n), m)] \\ _M. F. Hasler_, Feb 16 2023
%o (Python) m=-1; [n for n in range(9999)if m < (m := max(A003016(n), m))] # _M. F. Hasler_, Feb 16 2023
%K nonn,more
%O 1,3
%A _Reinhard Zumkeller_, Mar 20 2008