A237452 Zero-based row index to irregular tables organized as successively larger square matrices.

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

Offset: 1

Author

Antti Karttunen, Feb 08 2014

Keywords

• nonn
• tabf

Comments

With sequences constructed of successively larger kxk square matrices (cf. A074279), a(n) will return the distance of n from the top edge of the matrix that n is located in, with 0 standing for the topmost row in that matrix (please see the Example section).

A237451 gives the corresponding column index.

A238013 and A121997 give these same row and column indices, but starting the numbering with index 1. - M. F. Hasler, Feb 17 2014

Examples

			This irregular table begins as:
0;
0,0;
1,1;
0,0,0;
1,1,1;
2,2,2;
0,0,0,0;
1,1,1,1;
2,2,2,2;
3,3,3,3;
0,0,0,0,0;
1,1,1,1,1;
2,2,2,2,2;
3,3,3,3,3;
4,4,4,4,4;
...
		

Links

• Antti Karttunen, Table of squares with sizes 1x1 .. 30x30, flattened

Crossrefs

Cf. A064866, A074279, A237451, A237265, A238013 and A121997.

Programs

Python

from sympy import integer_nthroot
def A237452(n): return (n-1-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6)//k # Chai Wah Wu, Nov 04 2024

Scheme

(define (A237452 n) (floor->exact (/ (-1+ (A064866 n)) (A074279 n))))

Formula

a(n) = floor((A064866(n)-1)/A074279(n)).

a(n) = A238013(n)-1. - M. F. Hasler, Feb 16 2014