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

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, 19, 22, 25, 29, 26, 23, 20, 20, 23, 26, 29, 33, 30, 27, 24, 21, 24, 27, 30, 33, 37, 34, 31, 28, 25, 25, 28, 31, 34, 37, 41, 38, 35, 32, 29, 26, 29, 32, 35, 38, 41, 45

Offset: 1

Author

Boris Putievskiy, Jan 15 2013

Keywords

• nonn
• tabl

Comments

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.

Examples

			The start of the sequence as table for general case:
  1........m+1..2*m+1..3*m+1..4*m+1..5*m+1..6*m+1 ...
  m+1......m+2..2*m+2..3*m+2..4*m+2..5*m+2..6*m+2 ...
  2*m+1..2*m+2..2*m+3..3*m+3..4*m+3..5*m+3..6*m+3 ...
  3*m+1..3*m+2..3*m+3..3*m+4..4*m+4..5*m+4..6*m+4 ...
  4*m+1..4*m+2..4*m+3..4*m+4..4*m+5..5*m+5..6*m+5 ...
  5*m+1..5*m+2..5*m+3..5*m+4..5*m+5..5*m+6..6*m+6 ...
  6*m+1..6*m+2..6*m+3..6*m+4..6*m+5..6*m+6..6*m+7 ...
  . . .
The start of the sequence as triangle array read by rows for general case:
  1;
  m+1,     m+1;
  2*m+1,   m+2, 2*m+1;
  3*m+1, 2*m+2, 2*m+2, 3*m+1;
  4*m+1, 3*m+2, 2*m+3, 3*m+2, 4*m+1;
  5*m+1, 4*m+2, 3*m+3, 2*m+4, 3*m+3, 4*m+2; 5*m+1;
  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;
  . . .
Row number r contains r numbers: r*m+1, (r-1)*m+2, ... (r-1)*m+2, r*m+1.
		

Links

• Boris Putievskiy, Rows n = 1..140 of triangle, flattened
• Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Crossrefs

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

Programs

Python

t=int((math.sqrt(8*n-7)-1)/2)
result=4*(t+1)+3*max(t*(t+1)/2-n,n-(t*t+3*t+4)/2)

Formula

For the general case

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

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

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

For m=4

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

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