A138496 Where record values occur in A003016.

0, 1, 10, 120, 3003

Offset

1,3

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)

Links

Table of n, a(n) for n=1..5.

Prog

(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

Crossrefs

Sequence in context: A122420 A069671 A138445 * A111482 A095870 A178928

Adjacent sequences: A138493 A138494 A138495 * A138497 A138498 A138499

Keyword

nonn,more

Author

Reinhard Zumkeller, Mar 20 2008

Status

approved