A333516 Irregular triangle read by rows in which row n lists the first A000217(n) terms of A002260, n >= 1.

1, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7, 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6

Offset: 1

Author

Andrew Slattery, Mar 25 2020

Keywords

• nonn
• easy
• tabf

Comments

a(n) equals the difference between n and the largest number less than n that can be expressed as the sum of the i-th triangular number and the j-th tetrahedral number for integers i < j.

Examples

			Triangle begins:
  1;
  1, 1, 2;
  1, 1, 2, 1, 2, 3;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6;
  1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6, 7;
  ...
		

Crossrefs

Row sums give A000292.

Right border gives A000027.

Cf. A000217, A002260, A124171.

Programs

Maple

T:= n-> seq([$1..i][], i=1..n):
seq(T(n), n=1..7);  # Alois P. Heinz, Apr 10 2020

Python

from math import comb, isqrt
from sympy import integer_nthroot
def A333516(n): return (r:=n-1-comb((m:=integer_nthroot(6*n,3)[0])+(n>comb(m+2,3))+1,3))-comb((k:=isqrt(m:=r+1<<1))+(m>k*(k+1)),2)+1 # Chai Wah Wu, Nov 10 2024

Formula

a(n) = A002260(A124171(n)).