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

%I A237452 #14 Nov 05 2024 12:18:39
%S A237452 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 A237452 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 A237452 2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5
%N A237452 Zero-based row index to irregular tables organized as successively larger square matrices.
%C A237452 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 A237452 A237451 gives the corresponding column index.
%C A237452 A238013 and A121997 give these same row and column indices, but starting the numbering with index 1. - _M. F. Hasler_, Feb 17 2014
%H A237452 Antti Karttunen, <a href="/A237452/b237452.txt">Table of squares with sizes 1x1 .. 30x30, flattened</a>
%F A237452 a(n) = floor((A064866(n)-1)/A074279(n)).
%F A237452 a(n) = A238013(n)-1. - _M. F. Hasler_, Feb 16 2014
%e A237452 This irregular table begins as:
%e A237452 0;
%e A237452 0,0;
%e A237452 1,1;
%e A237452 0,0,0;
%e A237452 1,1,1;
%e A237452 2,2,2;
%e A237452 0,0,0,0;
%e A237452 1,1,1,1;
%e A237452 2,2,2,2;
%e A237452 3,3,3,3;
%e A237452 0,0,0,0,0;
%e A237452 1,1,1,1,1;
%e A237452 2,2,2,2,2;
%e A237452 3,3,3,3,3;
%e A237452 4,4,4,4,4;
%e A237452 ...
%o A237452 (Scheme) (define (A237452 n) (floor->exact (/ (-1+ (A064866 n)) (A074279 n))))
%o A237452 (Python)
%o A237452 from sympy import integer_nthroot
%o A237452 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 A237452 Cf. A064866, A074279, A237451, A237265, A238013 and A121997.
%K A237452 nonn,tabf
%O A237452 1,12
%A A237452 _Antti Karttunen_, Feb 08 2014