A138496 Where record values occur in A003016.

0, 1, 10, 120, 3003

Offset: 1

Author

Reinhard Zumkeller, Mar 20 2008

Keywords

• nonn
• more

Comments

It appears that the record values are 0, 3, 4, 6, 8, 10, 12, ...

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)

Programs

PARI

m=-1; [n | n<-[0..9999], m < m = max(A003016(n), m)] \\ M. F. Hasler, Feb 16 2023

Python

m=-1; [n for n in range(9999)if m < (m := max(A003016(n), m))] # M. F. Hasler, Feb 16 2023