A213921 - OEIS

%I #25 Nov 05 2025 15:22:22

%S 1,2,3,5,4,7,10,8,9,13,17,14,6,16,21,26,22,11,12,25,31,37,32,18,15,20,

%T 36,43,50,44,27,23,24,30,49,57,65,58,38,33,19,35,42,64,73,82,74,51,45,

%U 28,29,48,56,81,91,101,92,66,59,39,34,41,63,72,100,111

%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 clockwise. Table T(n,k) read by antidiagonals.

%C A 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). Enumeration table T(n,k) is 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(2,n), T(4,n), ... T(n,4), T(n,2);

%C ...

%H Boris Putievskiy, <a href="/A213921/b213921.txt">Rows n = 1..140 of triangle, flattened</a>

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

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/PairingFunction.html">MathWorld: Pairing functions</a>

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

%F As a table:

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

%F T(n,k) = k*k - ((n mod 2)+1)*k + floor((n+3)/2), if n<=k.

%F As a linear sequence:

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

%F a(n) = j*j - ((i mod 2)+1)*j + floor((i+3)/2), 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   8  14  22  32 ...

%e    7   9   6  11  18  27 ...

%e   13  16  12  15  23  33 ...

%e   21  25  20  24  19  28 ...

%e   31  36  30  35  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,  8,  9, 13;

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

%e   26, 22, 11, 12, 25, 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-(j%2)*i+2-int((j+2)/2)

%o else:

%o    result=j*j-((i%2)+1)*j + int((i+3)/2)

%Y Cf. A060734, A060736; table T(n,k) contains: in rows A002522, A014206, A059100, A027688, A117950, A027689, A087475, A027690, A117951, A027691, A114949, A027692, A117619; in columns A002061, A000290, A002378, A005563, A028387, A008865, A028552, A028872, A014209, A028347, A028875.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Mar 05 2013