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

1,2

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.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..11480 (result of A000217(1)..A000217(40).)

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 22.

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

Mathematica

Flatten@ Array[ConstantArray[#, #] & @* PolygonalNumber, 7] (* Michael De Vlieger, Nov 06 2025 *)

Prog

(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

Crossrefs

Cf. A000217, A002024, A072649, A074279.

Sequence in context: A112669 A098529 A133774 * A073080 A171601 A057944

Adjacent sequences: A108578 A108579 A108580 * A108582 A108583 A108584

Keyword

easy,nonn

Author

Jonathan Vos Post, Jul 25 2005

Status

approved