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A206772 Table T(n,k)=max{4*n+k-4,n+4*k-4} n, k > 0, read by antidiagonals.

%I A206772 #29 Nov 29 2023 06:57:58
%S A206772 1,5,5,9,6,9,13,10,10,13,17,14,11,14,17,21,18,15,15,18,21,25,22,19,16,
%T A206772 19,22,25,29,26,23,20,20,23,26,29,33,30,27,24,21,24,27,30,33,37,34,31,
%U A206772 28,25,25,28,31,34,37,41,38,35,32,29,26,29,32,35,38,41,45
%N A206772 Table T(n,k)=max{4*n+k-4,n+4*k-4} n, k > 0, read by antidiagonals.
%C A206772 In general, let m be natural number. Table T(n,k)=max{m*n+k-m,n+m*k-m}. For m=1 the result is A002024, for m=2 the result is A204004, for m=3  the result is A204008. This sequence is result for m=4.
%H A206772 Boris Putievskiy, <a href="/A206772/b206772.txt">Rows n = 1..140 of triangle, flattened</a>
%H A206772 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A206772 For the general case
%F A206772 a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.
%F A206772 a(n) = m*(t+1) + (m-1)*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}
%F A206772 where t=floor((-1+sqrt(8*n-7))/2).
%F A206772 For m=4
%F A206772 a(n) = 4*(t+1) + 3*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}
%F A206772 where t=floor((-1+sqrt(8*n-7))/2).
%e A206772 The start of the sequence as table for general case:
%e A206772   1........m+1..2*m+1..3*m+1..4*m+1..5*m+1..6*m+1 ...
%e A206772   m+1......m+2..2*m+2..3*m+2..4*m+2..5*m+2..6*m+2 ...
%e A206772   2*m+1..2*m+2..2*m+3..3*m+3..4*m+3..5*m+3..6*m+3 ...
%e A206772   3*m+1..3*m+2..3*m+3..3*m+4..4*m+4..5*m+4..6*m+4 ...
%e A206772   4*m+1..4*m+2..4*m+3..4*m+4..4*m+5..5*m+5..6*m+5 ...
%e A206772   5*m+1..5*m+2..5*m+3..5*m+4..5*m+5..5*m+6..6*m+6 ...
%e A206772   6*m+1..6*m+2..6*m+3..6*m+4..6*m+5..6*m+6..6*m+7 ...
%e A206772   . . .
%e A206772 The start of the sequence as triangle array read by rows for general case:
%e A206772   1;
%e A206772   m+1,     m+1;
%e A206772   2*m+1,   m+2, 2*m+1;
%e A206772   3*m+1, 2*m+2, 2*m+2, 3*m+1;
%e A206772   4*m+1, 3*m+2, 2*m+3, 3*m+2, 4*m+1;
%e A206772   5*m+1, 4*m+2, 3*m+3, 2*m+4, 3*m+3, 4*m+2; 5*m+1;
%e A206772   6*m+1, 5*m+2, 4*m+3, 3*m+4, 2*m+5, 3*m+4, 4*m+3; 5*m+2, 6*m+1;
%e A206772   . . .
%e A206772 Row number r contains r numbers: r*m+1, (r-1)*m+2, ... (r-1)*m+2, r*m+1.
%o A206772 (Python)
%o A206772 t=int((math.sqrt(8*n-7)-1)/2)
%o A206772 result=4*(t+1)+3*max(t*(t+1)/2-n,n-(t*t+3*t+4)/2)
%Y A206772 Cf. A002024, A204004, A204008, A002260, A004736.
%K A206772 nonn,tabl
%O A206772 1,2
%A A206772 _Boris Putievskiy_, Jan 15 2013