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%I #18 Sep 08 2022 08:45:10
%S 1,3,2,8,5,4,20,13,10,6,49,32,25,15,7,119,78,61,37,17,9,288,189,148,
%T 90,42,22,11,696,457,358,218,102,54,27,12,1681,1104,865,527,247,131,
%U 66,29,14,4059,2666,2089,1273,597,317,160,71,34,16,9800,6437,5044,3074
%N 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).
%C 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
%F T(n,k+1) = 2*T(n,k) + T(n,k-1) + 1 for n>=0, k>=1.
%e Table begins:
%e 1 3 8 20 49 119 288 ...
%e 2 5 13 32 78 189 457 ...
%e 4 10 25 61 148 358 865 ...
%e 6 15 37 90 218 527 1273 ...
%e 7 17 42 102 247 597 1442 ...
%e 9 22 54 131 317 766 1850 ...
%e 11 27 66 160 387 935 2258 ...
%e 12 29 71 172 416 1005 2427 ...
%e 14 34 83 201 486 1174 2835 ...
%e 16 39 95 230 556 1343 3243 ...
%e 18 44 107 259 626 1512 3651 ...
%e 19 46 112 271 655 1582 3820 ...
%e 21 51 124 300 725 1751 4228 ...
%e 23 56 136 329 795 1920 4636 ...
%e 24 58 141 341 824 1990 4805 ...
%t (See Comments.)
%o (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
%Y Cf. A083088 (first column), A048739 (first row), A083090 (diagonal), A083091 (antidiagonal sums), A083044, A083047, A083050.
%K nonn,tabl
%O 0,2
%A _Paul D. Hanna_, Apr 21 2003
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