A108581 Positive triangular numbers repeated their own number of times.

1, 3, 3, 3, 6, 6, 6, 6, 6, 6, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 15, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 21, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28, 28

Offset: 1

Author

Jonathan Vos Post, Jul 25 2005

Keywords

• easy
• nonn

Comments

In terms of the repetition convolution operator #, where (sequence A) # (sequence B) = the sequence consisting of A(n) copies of B(n), then this sequence is the repetition self-convolution (A000217(n)-0) # (A000217(n)-0). Over the set of positive infinite integer sequences, # gives a nonassociative noncommutative groupoid with a left identity (A000012) but no right identity, where the left identity is also a right nullifier and idempotent.

Crossrefs

Cf. A000217, A002024, A072649, A074279.

Programs

Python

from sympy import integer_nthroot
def A108581(n): return (r:=(m:=integer_nthroot(k:=6*n,3)[0])+(k>m*(m+1)*(m+2)))*(r+1)>>1 # Chai Wah Wu, Nov 07 2024

Formula

a(1) = 1, for n>1: a(A000217(n)-1) = a(A000217(n)) = ... = a(A000217(n+1)-2) = A000217(n).

a(n) = A000217(m+1) if 6n>m(m+1)(m+2) and a(n) = A000217(m) otherwise where m = floor((6n)^(1/3)). - Chai Wah Wu, Nov 07 2024