A352269 -id:A352269 - cp's OEIS frontend

A351846 Irregular triangle read by rows: T(n,k), n >= 0, k >= 0, in which n appears 4*n + 1 times in row n.

0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6

Offset: 0

Omar E. Pol, Feb 21 2022

a(n) is the number of hexagonal numbers A000384 less than or equal to n, not counting 0 as hexagonal.

This sequence is related to hexagonal numbers as A003056 is related to triangular numbers (or generalized hexagonal numbers) A000217.

			Triangle begins:
  0;
  1, 1, 1, 1, 1;
  2, 2, 2, 2, 2, 2, 2, 2, 2;
  3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
  4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;
  5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5;
  6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6;
  ...

Row sums give A007742.
Row n has length A016813(n).
Column 0 gives A001477, the same as the right border.
Nonzero terms give the row lengths of the triangles A347263, A347529, A351819, A351824, A352269, A352499.
Cf. A000217, A000384, A003056.

Table[PadRight[{},4n+1,n],{n,0,7}]//Flatten (* Harvey P. Dale, Jun 04 2023 *)

a(n) = floor((sqrt(8*n + 1) + 1)/4). - Ridouane Oudra, Apr 09 2023

A341309 Sum of odd divisors of n that are <= A003056(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 9, 1, 1, 4, 1, 6, 4, 1, 1, 4, 6, 1, 4, 8, 1, 9, 1, 1, 4, 1, 13, 4, 1, 1, 4, 6, 1, 11, 1, 1, 18, 1, 1, 4, 8, 6, 4, 1, 1, 13, 6, 8, 4, 1, 1, 9, 1, 1, 20, 1, 6, 15, 1, 1, 4, 13, 1, 13, 1, 1, 9, 1, 19, 4, 1, 6, 13, 1, 1, 11, 6, 1, 4, 12, 1, 18, 21

Offset: 1

N. J. A. Sloane, Feb 14 2021

nonn

Conjecture 1: a(n) is also the total number de parts in all partitions of n into an odd number of consecutive parts. - Omar E. Pol, Mar 16 2022

Conjecture 2: row sums of A352269. - Omar E. Pol, Mar 18 2022

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A000593, A003056, A082647, A131576, A333807, A341310.

A341309[n_]:=With[{t=Floor[(Sqrt[8n+1]-1)/2]},DivisorSum[n,#&,OddQ[#]&&#<=t&]];
Array[A341309,100] (* Paolo Xausa, Mar 25 2023 *)

a(n) = my(m=n>>valuation(n, 2), s=(sqrtint(8*n+1)-1)\2); sumdiv(m, d, if (d <= s, d)); \\ Michel Marcus, Mar 25 2023

a(n) = A204217(n) - A352446(n), conjectured. - Omar E. Pol, Mar 16 2022

cp's OEIS Frontend

A351846 Irregular triangle read by rows: T(n,k), n >= 0, k >= 0, in which n appears 4*n + 1 times in row n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula