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Zero-based row index to irregular tables organized as successively larger square matrices.
6

%I #14 Nov 05 2024 12:18:39

%S 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,

%T 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,

%U 2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5

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

%C 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).

%C A237451 gives the corresponding column index.

%C A238013 and A121997 give these same row and column indices, but starting the numbering with index 1. - _M. F. Hasler_, Feb 17 2014

%H Antti Karttunen, <a href="/A237452/b237452.txt">Table of squares with sizes 1x1 .. 30x30, flattened</a>

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

%F a(n) = A238013(n)-1. - _M. F. Hasler_, Feb 16 2014

%e This irregular table begins as:

%e 0;

%e 0,0;

%e 1,1;

%e 0,0,0;

%e 1,1,1;

%e 2,2,2;

%e 0,0,0,0;

%e 1,1,1,1;

%e 2,2,2,2;

%e 3,3,3,3;

%e 0,0,0,0,0;

%e 1,1,1,1,1;

%e 2,2,2,2,2;

%e 3,3,3,3,3;

%e 4,4,4,4,4;

%e ...

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

%o (Python)

%o from sympy import integer_nthroot

%o 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

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

%K nonn,tabf

%O 1,12

%A _Antti Karttunen_, Feb 08 2014