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

%I A237451 #17 Nov 05 2024 03:12:03
%S A237451 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,
%T A237451 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,
%U A237451 1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5
%N A237451 Zero-based column index to irregular tables organized as successively larger square matrices.
%C A237451 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).
%C A237451 A237452 gives the corresponding row index.
%C A237451 A238013 and A121997 give these same row and column indices, starting the numbering with index 1. - M. F. Hasler, Feb 17 2014
%H A237451 Antti Karttunen, <a href="/A237451/b237451.txt">Table of squares with sizes 1x1 .. 30x30, flattened</a>
%F A237451 a(n) = (A064866(n)-1) modulo A074279(n).
%F A237451 a(n) = A121997(n)-1. - _M. F. Hasler_, Feb 16 2014
%e A237451 This irregular table begins as:
%e A237451 0;
%e A237451 0,1;
%e A237451 0,1;
%e A237451 0,1,2;
%e A237451 0,1,2;
%e A237451 0,1,2;
%e A237451 0,1,2,3;
%e A237451 0,1,2,3;
%e A237451 0,1,2,3;
%e A237451 0,1,2,3;
%e A237451 0,1,2,3,4;
%e A237451 0,1,2,3,4;
%e A237451 0,1,2,3,4;
%e A237451 0,1,2,3,4;
%e A237451 0,1,2,3,4;...
%o A237451 (Scheme) (define (A237451 n) (modulo (-1+ (A064866 n)) (A074279 n)))
%o A237451 (Python)
%o A237451 from sympy import integer_nthroot
%o A237451 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
%Y A237451 Cf. A002262, A064866, A074279, A237452, A237265.
%K A237451 nonn,tabf,easy
%O A237451 1,8
%A A237451 _Antti Karttunen_, Feb 08 2014