A214870 - OEIS

%I #27 Nov 29 2023 08:15:40

%S 1,2,3,5,4,7,10,9,8,13,17,16,6,14,21,26,25,11,12,22,31,37,36,18,15,20,

%T 32,43,50,49,27,24,23,30,44,57,65,64,38,35,19,33,42,58,73,82,81,51,48,

%U 28,29,45,56,74,91,101,100,66,63,39,34,41,59,72,92,111

%N Natural numbers placed in table T(n,k) layer by layer. The order of placement: at the beginning filled odd places of layer clockwise, next - even places counterclockwise. T(n,k) read by antidiagonals.

%C Permutation of the natural numbers.

%C a(n) is a pairing function: a function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} is the set of integer positive numbers.

%C Layer is pair of sides of square from T(1,n) to T(n,n) and from T(n,n) to T(n,1).

%C Enumeration table T(n,k) layer by layer. The order of the list:

%C   T(1,1)=1;

%C   T(1,2), T(2,1), T(2,2);

%C   . . .

%C   T(1,n), T(3,n), ... T(n,3), T(n,1); T(n,2), T(n,4), ... T(4,n), T(2,n);

%C   . . .

%H Boris Putievskiy, <a href="/A214870/b214870.txt">Rows n = 1..140 of triangle, flattened</a>

%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations [of] Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PairingFunction.html">Pairing functions</a>

%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>

%F As table

%F T(n,k) = k*k-2*(n mod 2)*k+(n mod 2)*((n+1)/2+1)+((n+1) mod 2)*(-n/2+1), if n<=k;

%F T(n,k) = n*n-n+(k mod 2)*(2-(k+1)/2)+((k+1) mod 2)*(k/2+1), if n>k.

%F As linear sequence

%F a(n) = j*j-2*(i mod 2)*j+(i mod 2)*((i+1)/2+1)+((i+1) mod 2)*(-i/2+1), if i<=j;

%F a(n) = i*i-i+(j mod 2)*(2-(j+1)/2)+((j+1) mod 2)*(j/2+1), 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).

%e The start of the sequence as table:

%e    1   2   5  10  17  26 ...

%e    3   4   9  16  25  36 ...

%e    7   8   6  11  18  27 ...

%e   13  14  12  15  24  35 ...

%e   21  22  20  23  19  28 ...

%e   31  32  30  33  29  34 ...

%e   ...

%e The start of the sequence as triangle array read by rows:

%e    1;

%e    2,  3;

%e    5,  4,  7;

%e   10,  9,  8, 13;

%e   17, 16,  6, 14, 21;

%e   26, 25, 11, 12, 22, 31;

%e   ...

%o (Python)

%o t=int((math.sqrt(8*n-7) - 1)/ 2)

%o i=n-t*(t+1)/2

%o j=(t*t+3*t+4)/2-n

%o if i > j:

%o    result=i*i-i+(j%2)*(2-(j+1)/2)+((j+1)%2)*(j/2+1)

%o else:

%o    result=j*j-2*(i%2)*j + (i%2)*((i+1)/2+1) + ((i+1)%2)*(-i/2+1)

%Y Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A002522, A000290, A059100, A005563, A117950, A008865, A087475, A028872, A117951, A028347, A114949, A028875, A117619, A028878, A189833, A028881, A189834, A028884, A114948, A028560, A189836; in columns A002061, A014206, A002378, A027688, A028387, A027689, A028552, A027690, A014209, A027691, A027692, A082111, A027693, A028557, A027694, A108195, A187710, A048058, A048840.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Mar 11 2013