A130296 Triangle read by rows: T[i,1]=i, T[i,j]=1 for 1 < j <= i = 1,2,3,...

1, 2, 1, 3, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 14, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Gary W. Adamson, May 20 2007

Keywords

• nonn
• tabl
• easy

Comments

This sequence was formerly named "reversals of A051340", but it is actually the truncation of A051340 to its lower left triangular part, re-indexed to start rows and columns with 1. - M. F. Hasler, Aug 15 2015

Examples

			First few rows of the triangle are:
1;
2, 1;
3, 1, 1;
4, 1, 1, 1;
5, 1, 1, 1, 1;
...
		

Crossrefs

Cf. A051340, A130297, A005408 (row sums).

Programs

PARI

A130296(i,j)=if(j==1,i,j<=i) \\ The sequence should not be defined for j>i but it is used in several places as infinite square matrix with upper right part equal to zero. - M. F. Hasler, Aug 15 2015

Python

from math import isqrt
def A130296(n): return comb((m:=isqrt(k:=n<<1))+(k>m*(m+1)),2)-comb((m2:=isqrt(k-2))+(k-2>m2*(m2+1)),2)+1 # Chai Wah Wu, Nov 09 2024

Formula

Truncation of A051340 to its lower left. By rows, "n" followed by (n-1) 1's. (1,2,3...) in the left border, all 1's in other columns.

A130296^2 = A130297.

a(n) = A004201(n) - A004201(n-1) for n>1. - Reinhard Zumkeller, Jul 16 2008