A237451 Zero-based column index to irregular tables organized as successively larger square matrices.

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

Offset: 1

Author

Antti Karttunen, Feb 08 2014

Keywords

• nonn
• tabf
• easy

Comments

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

A237452 gives the corresponding row index.

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

Examples

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

Links

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

Crossrefs

Cf. A002262, A064866, A074279, A237452, A237265.

Programs

Python

from sympy import integer_nthroot
def A237451(n): return (n-(k:=(m:=integer_nthroot(3*n,3)[0])+(6*n>m*(m+1)*((m<<1)+1)))*(k-1)*((k<<1)-1)//6-1)%k # Chai Wah Wu, Nov 04 2024

Scheme

(define (A237451 n) (modulo (-1+ (A064866 n)) (A074279 n)))

Formula

a(n) = (A064866(n)-1) modulo A074279(n).

a(n) = A121997(n)-1. - M. F. Hasler, Feb 16 2014