A216252 A213196 as table read layer by layer clockwise.
1, 4, 5, 2, 3, 7, 10, 8, 6, 11, 9, 17, 20, 23, 14, 12, 13, 16, 26, 38, 43, 39, 21, 24, 15, 22, 25, 30, 42, 58, 63, 48, 35, 31, 27, 18, 19, 29, 34, 57, 53, 69, 76, 70, 64, 49, 36, 32, 28, 37, 33, 47, 52, 81, 75, 95, 102, 109, 88, 82, 54, 59, 44, 40, 41, 46, 62
Offset: 1
Examples
The start of the sequence as table: 1....4...3..11..13... 2....5...7...9..16... 6....8..10..17..26... 12..14..23..20..38... 15..24..21..39..43... . . . The start of the sequence as triangular array read by rows: 1; 4,5,2; 3,7,10,8,6; 11,9,17,20,23,14,12; 13,16,26,38,43,39,21,24,15; . . . Row number r contains 2*r-1 numbers.
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, MathWorld: Pairing functions
- Index entries for sequences that are permutations of the natural numbers
Programs
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Python
t=int((math.sqrt(n-1)))+1 i=min(t,n-(t-1)**2) j=min(t,t**2-n+1) m1=(3*i+j-1-(-1)**i+(i+j-2)*(-1)**(i+j))/4 m2=((1+(-1)**i)*((1+(-1)**j)*2*int((j+2)/4)-(-1+(-1)**j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)**i)*((1+(-1)**j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)**j)*(1+2*int(j/4))))/4 m=(m1+m2-1)*(m1+m2-2)/2+m1
Formula
a(n) = (m1+m2-1)*(m1+m2-2)/2+m1, where m1=(3*i+j-1-(-1)^i+(i+j-2)*(-1)^(i+j))/4, m2=((1+(-1)^i)*((1+(-1)^j)*2*int((j+2)/4)-(-1+(-1)^j)*(2*int((i+4)/4)+2*int(j/2)))-(-1+(-1)^i)*((1+(-1)^j)*(1+2*int(i/4)+2*int(j/2))-(-1+(-1)^j)*(1+2*int(j/4))))/4, i=min(t; n-(t-1)^2), j=min(t; t^2-n+1), t=floor(sqrt(n-1))+1.
Comments