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

1,2

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.

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.

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).

Example

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.

Prog

(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)

Crossrefs

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

Sequence in context: A030798 A331502 A021951 * A200679 A124175 A168277

Adjacent sequences: A206769 A206770 A206771 * A206773 A206774 A206775

Keyword

nonn,tabl

Author

Boris Putievskiy, Jan 15 2013

Status

approved