A214871 Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.
1, 3, 4, 6, 2, 7, 11, 8, 9, 12, 18, 13, 5, 14, 19, 27, 20, 15, 16, 21, 28, 38, 29, 22, 10, 23, 30, 39, 51, 40, 31, 24, 25, 32, 41, 52, 66, 53, 42, 33, 17, 34, 43, 54, 67, 83, 68, 55, 44, 35, 36, 45, 56, 69, 84, 102, 85, 70, 57, 46, 26, 47, 58, 71, 86, 103, 123
Offset: 1
Examples
The start of the sequence as table: 1....3...6..11..18..27... 4....2...8..13..20..29... 7....9...5..15..22..31... 12..14..16..10..24..33... 19..21..23..25..17..35... 28..30..32..34..36..26... . . . The start of the sequence as triangle array read by rows: 1; 3,4; 6,2,7; 11,8,9,12; 18,13,5,14,19; 27,20,15,16,21,28; . . .
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.
- Eric Weisstein's World of Mathematics, Pairing functions
- Index entries for sequences that are permutations of the natural numbers
Crossrefs
Programs
-
Python
t=int((math.sqrt(8*n-7) - 1)/ 2) i=n-t*(t+1)/2 j=(t*t+3*t+4)/2-n if i == j: result=(i-1)**2+1 if i > j: result=(i-1)**2+2*j+1 if i < j: result=(j-1)**2+2*i
Formula
As table
T(n,k) = (n-1)^2+1, if n=k;
T(n,k) = (n-1)^2+2*k+1, if n>k;
T(n,k) = (k-1)^2+2*n, if n
As linear sequence
a(n) = (i-1)^2+1, if i=j;
a(n) = (i-1)^2+2*j+1, if i>j;
a(n) = (j-1)^2+2*i, if i>j; where i=n-t*(t+1)/2, j=(t*t+3*t+4)/2-n, t=floor((-1+sqrt(8*n-7))/2).
Comments