cp's OEIS Frontend

Row sums of A004248, A009998, A009999.

First differences are in A047970.

First differences of A103439.

Antidiagonal sums of array A003992.

a(n-1), for n>=1, is the number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k)<=1+max(prefix) for k>=1, that are simultaneously projections as maps f: [n] -> [n] where f(x)<=x and f(f(x))=f(x); see example and the two comments (Arndt, Apr 30 2011 Jan 04 2013) in A000110. - Joerg Arndt, Mar 07 2015

Number of finite sequences s of length n+1 whose discriminator sequence is s itself. Here the discriminator sequence of s is the one where the n-th term (n>=1) is the least positive integer k such that the first n terms are pairwise incongruent, modulo k. - Jeffrey Shallit, May 17 2016

From Gus Wiseman, Jan 08 2019: (Start)

Also the number of set partitions of {1,...,n+1} whose minima form an initial interval of positive integers. For example, the a(3) = 9 set partitions are:

{{1},{2},{3},{4}}

{{1},{2},{3,4}}

{{1},{2,4},{3}}

{{1,4},{2},{3}}

{{1},{2,3,4}}

{{1,3},{2,4}}

{{1,4},{2,3}}

{{1,3,4},{2}}

{{1,2,3,4}}

Missing from this list are:

{{1},{2,3},{4}}

{{1,2},{3},{4}}

{{1,3},{2},{4}}

{{1,2},{3,4}}

{{1,2,3},{4}}

{{1,2,4},{3}}

(End)

a(n) is the number of m-tuples of nonnegative integers less than or equal to n-m (including the "0-tuple"). - Mathew Englander, Apr 11 2021

From Lara Pudwell, Oct 23 2008: (Start)

A permutation p avoids a pattern q if it has no subsequence that is order-isomorphic to q. For example, p avoids the pattern 132 if it has no subsequence abc with a < c < b.

Barred pattern avoidance considers permutations that avoid a pattern except in a special case. Given a barred pattern q, we may form two patterns, q1 = the sequence of unbarred letters of q and q2 = the sequence of all letters of q.

A permutation p avoids barred pattern q if every instance of q1 in p is embedded in a copy of q2 in p. In other words, p avoids q1, except in the special case that a copy of q1 is a subsequence of a copy of q2.

For example, if q=5{bar 1}32{bar 4}, then q1=532 and q2 = 51324. p avoids q if every for decreasing subsequence acd of length 3 in p, one can find letters b and e so that the subsequence abcde of p has b < d < c < e < a. (End)

Number of ordered factorizations over the Gaussian polynomials.

Apparently, also the number of permutations in S_n avoiding {bar 3}{bar 1}542 (i.e., every occurrence of 542 is contained in an occurrence of a 31542). - Lara Pudwell, Apr 25 2008

With offset 1, apparently the number of sequences {b(m)} of length n of positive integers with b(1) = 1 and, for all m > 1, b(m) <= max{b(m-1) + 1, max{b(i) | 1 <= i <= m - 1}}. This sequence begins 1, 2, 5, 14, 43, 144, 523, 2048, 8597, 38486. The term 144 counts the length 6 sequence 1, 2, 3, 1, 1, 3, for instance. Contrast with the families of sequences discussed in Franklin T. Adams-Watters's comment in A005425. - Rick L. Shepherd, Jan 01 2015

a(n-1) for n >= 1 is the number of length-n restricted growth strings (RGS) [s(0), s(1), ..., s(n-1)] with s(0)=0 and s(k) <= the number of fixed points in the prefix, see example. - Joerg Arndt, Mar 08 2015

Number of sequences (e(1), ..., e(n+1)), 0 <= e(i) < i, such that there is no triple i < j < k with e(i) != e(j) = e(k). [Martinez and Savage, 2.15] - Eric M. Schmidt, Jul 17 2017

a(n) counts all positive-integer m-tuples whose maximum is n-m+2. - Mathew Englander, Feb 28 2021

a(n) counts the cyclic permutations of [n+2] that avoid the vincular pattern 12-3-4, i.e., the pattern 1234 where the 1 and 2 are required to be adjacent. - Rupert Li, Jul 27 2021

(n-th term) = (n-th term of A002260)^(n-th term of A004736). Both A002260 and A004736 are related to A002024. - Robert A. Stump (bee_ess107(AT)yahoo.com), Aug 29 2002

The maximal absolute difference is assumed to be zero if there are fewer than two blocks.

T(n,k) is defined for all n,k >= 0. The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0.

Sum of antidiagonals is A003101(n) for n>0. - Alford Arnold, Jan 14 2007

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. Since the parity of the triangular numbers cycles odd, odd, even, even; the parity of this sequence cycles odd, even, even, even. The smallest prime in this sequence is a(5) = 127601. What is the next prime? What is the first triangular value?

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest prime in this sequence is a(2) = 5. What is the next prime? What is the first square value after 1?

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest primes in this (always odd) sequence are a(1) = 3, a(3) = 41 and a(5) = 543. What is the next prime?

A003101 is the ascending descending base exponent transform of natural numbers A000027. The ascending descending base exponent transform applied to the Fibonacci numbers is A113122; applied to the tribonacci numbers is A113153; applied to the Lucas numbers is A113154. The smallest primes in this (always odd) sequence are a(2) = 3 and a(3) = 11. What is the next prime? Is there a nontrivial power after a(4) = 5^3?