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A206772 Table T(n,k)=max{4*n+k-4,n+4*k-4} n, k > 0, read by antidiagonals. 3


%S 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 19,22,25,29,26,23,20,20,23,26,29,33,30,27,24,21,24,27,30,33,37,34,31,

%U 28,25,25,28,31,34,37,41,38,35,32,29,26,29,32,35,38,41,45

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

%C 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 Boris Putievskiy, <a href="/A206772/b206772.txt">Rows n = 1..140 of triangle, flattened</a>

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO]

%F For the general case

%F a(n) = m*A002024(n) + (m-1)*max{-A002260(n),-A004736(n)}.

%F a(n) = m*(t+1) + (m-1)*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}

%F where t=floor((-1+sqrt(8*n-7))/2).

%F For m=4

%F a(n) = 4*(t+1) + 3*max{t*(t+1)/2-n,n-(t*t+3*t+4)/2}

%F where t=floor((-1+sqrt(8*n-7))/2).

%e The start of the sequence as table for general case:

%e 1........m+1..2*m+1..3*m+1..4*m+1..5*m+1..6*m+1 ...

%e m+1......m+2..2*m+2..3*m+2..4*m+2..5*m+2..6*m+2 ...

%e 2*m+1..2*m+2..2*m+3..3*m+3..4*m+3..5*m+3..6*m+3 ...

%e 3*m+1..3*m+2..3*m+3..3*m+4..4*m+4..5*m+4..6*m+4 ...

%e 4*m+1..4*m+2..4*m+3..4*m+4..4*m+5..5*m+5..6*m+5 ...

%e 5*m+1..5*m+2..5*m+3..5*m+4..5*m+5..5*m+6..6*m+6 ...

%e 6*m+1..6*m+2..6*m+3..6*m+4..6*m+5..6*m+6..6*m+7 ...

%e . . .

%e The start of the sequence as triangle array read by rows for general case:

%e 1;

%e m+1, m+1;

%e 2*m+1, m+2, 2*m+1;

%e 3*m+1, 2*m+2, 2*m+2, 3*m+1;

%e 4*m+1, 3*m+2, 2*m+3, 3*m+2, 4*m+1;

%e 5*m+1, 4*m+2, 3*m+3, 2*m+4, 3*m+3, 4*m+2; 5*m+1;

%e 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 . . .

%e Row number r contains r numbers: r*m+1, (r-1)*m+2, ... (r-1)*m+2, r*m+1.

%o (Python)

%o t=int((math.sqrt(8*n-7)-1)/2)

%o result=4*(t+1)+3*max(t*(t+1)/2-n,n-(t*t+3*t+4)/2)

%Y Cf. A002024, A204004, A204008, A002260, A004736.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Jan 15 2013

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Last modified March 23 02:42 EDT 2019. Contains 321422 sequences. (Running on oeis4.)