A083087 Square table read by antidiagonals which forms a permutation of the natural numbers: T(n,0) = floor(n*x/(x-1))+1, T(n,k+1) = ceiling(x*T(n,k)), for n>=0, k>=0, where x = 1 + sqrt(2).

1, 3, 2, 8, 5, 4, 20, 13, 10, 6, 49, 32, 25, 15, 7, 119, 78, 61, 37, 17, 9, 288, 189, 148, 90, 42, 22, 11, 696, 457, 358, 218, 102, 54, 27, 12, 1681, 1104, 865, 527, 247, 131, 66, 29, 14, 4059, 2666, 2089, 1273, 597, 317, 160, 71, 34, 16, 9800, 6437, 5044, 3074

Offset: 0

Author

Paul D. Hanna, Apr 21 2003

Keywords

• nonn
• tabl

Comments

The array in A083087 is the dispersion of the sequence given floor(n+1+n*sqrt(2)). The Mathematica program at A191438 generates A083087 using f[n_]:=Floor[n*x+n+1] instead of f[n_]:=Floor[n*x+n]. - Clark Kimberling, Jun 04 2011

Examples

			Table begins:
   1  3   8  20  49  119  288 ...
   2  5  13  32  78  189  457 ...
   4 10  25  61 148  358  865 ...
   6 15  37  90 218  527 1273 ...
   7 17  42 102 247  597 1442 ...
   9 22  54 131 317  766 1850 ...
  11 27  66 160 387  935 2258 ...
  12 29  71 172 416 1005 2427 ...
  14 34  83 201 486 1174 2835 ...
  16 39  95 230 556 1343 3243 ...
  18 44 107 259 626 1512 3651 ...
  19 46 112 271 655 1582 3820 ...
  21 51 124 300 725 1751 4228 ...
  23 56 136 329 795 1920 4636 ...
  24 58 141 341 824 1990 4805 ...
		

Crossrefs

Cf. A083088 (first column), A048739 (first row), A083090 (diagonal), A083091 (antidiagonal sums), A083044, A083047, A083050.

Programs

Magma

z:=10; x:=1+Sqrt(2); S:=[]; for n in [0..z] do for k in [0..n] do if n-k eq 0 then Append(~S, Floor(n*x/(x-1))+1); else Append(~S, Ceiling(x*S[k+1+(n*(n-1) div 2)])); end if; end for; end for; S; // Klaus Brockhaus, Jan 04 2011

Mathematica

(See Comments.)

Formula

T(n,k+1) = 2*T(n,k) + T(n,k-1) + 1 for n>=0, k>=1.