A216249 - OEIS

%I #27 Nov 20 2019 09:33:37

%S 1,3,2,4,5,6,8,7,12,11,9,10,13,14,15,17,16,21,20,25,24,18,19,22,23,26,

%T 27,28,30,29,34,33,38,37,42,41,31,32,35,36,39,40,43,44,45,47,46,51,50,

%U 55,54,59,58,63,62,48,49,52,53,56,57,60,61,64,65,66,68,67,72,71,76,75,80,79,84,83,88,87

%N T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1; T(n,k)=((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2, else. Table T(n,k) read by antidiagonals; n , k > 0.

%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 Enumeration table T(n,k). Let m be natural number. The order of the list:

%C T(1,1)=1;

%C T(2,1), T(1,2), T(1,3), T(2,2), T(3,1);

%C . . .

%C T(2,2*m-1), T(1,2*m), T(1,2*m+1), T(2,2*m), T(2*m-3,4), ... T(2*m,1), T(2*m-1,2), T(2*m-1,3), T(2*m,2), T(2*m+1,1);

%C . . .

%C Movement along two adjacent antidiagonals - step to the northeast, step to the east, step to the southwest, 3 steps to the west, 2 steps to the south and so on.

%C The length of each step is 1.

%H Boris Putievskiy, <a href="/A216249/b216249.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 a table:

%F T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2-2, if k=1 and (n mod 2)=1;

%F T(n,k) = ((n+k)^2-4*k+3-2*(-1)^n+(-1)^k-(n+k-4)*(-1)^(n+k))/2,   else.

%F As a linear sequence:

%F a(n) = ((t+2)^2-4*j+3-2*(-1)^i+(-1)^j-(t-2)*(-1)^t)/2-2, if j=1 and (i mod 2)=1;

%F a(n) = ((t+2)^2-4*j+3-2*(-1)^i+(-1)^j-(t-2)*(-1)^t)/2,   else; 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   3  4    8   9  17  18...

%e    2   5  7   10  16  19  29...

%e    6  12  13  21  22  34  35...

%e   11  14  20  23  33  36  50...

%e   15  25  26  38  39  55  56...

%e   24  27  37  40  54  57  75...

%e   28  42  43  59  60  80  81...

%e   ...

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

%e    1;

%e    3,  2;

%e    4,  5,  6;

%e    8,  7, 12, 11;

%e    9, 10, 13, 14, 15;

%e   17, 16, 21, 20, 25, 24;

%e   18, 19, 22, 23, 26, 27, 28;

%e   ...

%e As an array read by rows, where the length of row number r is 4*r-3:

%e First 2*r-2 numbers are from the row number 2*r-2 of triangle array, located above.

%e Last  2*r-1 numbers are from the row number 2*r-1 of triangle array, located above.

%e   1;

%e   3,   2,   4,   5,   6;

%e   8,   7,  12,  11,   9,  10,  13,  14,  15;

%e   17, 16,  21,  20,  25,  24,  18,  19,  22,  23,  26,  27,  28;

%e   ...

%e Row number r contains permutation of the 4*r-3 numbers from 2*r*r-5*r+4 to 2*r*r-r:

%e 2*r*r-5*r+5, 2*r*r-5*r+4, ...2*r*r-r-1, 2*r*r-r.

%t T[n_, k_] := ((n+k)^2 - 4k + 3 - 2(-1)^n + (-1)^k - (n+k-4)(-1)^(n+k))/2 - 2Boole[k == 1 && OddQ[n]];

%t Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Nov 20 2019 *)

%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 result=((t+2)**2-4*j+3+(-1)**j-2*(-1)**i-(t-2)*(-1)**t)/2

%o if j==1 and (i%2)==1:

%o    result=result-2

%Y Cf. A213205, A213171, A213197, A210521; table T(n,k) contains: in rows A100037, A033816, A130883, A100039, A100038; in columns A000384, A071355, A091823, A014106.

%K nonn,tabl

%O 1,2

%A _Boris Putievskiy_, Mar 14 2013