|
%I #20 Nov 29 2023 08:15:22
%S 1,3,4,6,2,7,11,8,9,12,18,13,5,14,19,27,20,15,16,21,28,38,29,22,10,23,
%T 30,39,51,40,31,24,25,32,41,52,66,53,42,33,17,34,43,54,67,83,68,55,44,
%U 35,36,45,56,69,84,102,85,70,57,46,26,47,58,71,86,103,123
%N Natural numbers placed in table T(n,k) layer by layer. The order of placement - T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1). Table T(n,k) read by antidiagonals.
%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(2,2), T(1,2), T(2,1);
%C . . .
%C T(n,n), T(1,n), T(n,1), T(2,n), T(n,2),...T(n-1,n), T(n,n-1);
%C . . .
%H Boris Putievskiy, <a href="/A214871/b214871.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) = (n-1)^2+1, if n=k;
%F T(n,k) = (n-1)^2+2*k+1, if n>k;
%F T(n,k) = (k-1)^2+2*n, if n<k.
%F As linear sequence
%F a(n) = (i-1)^2+1, if i=j;
%F a(n) = (i-1)^2+2*j+1, if i>j;
%F a(n) = (j-1)^2+2*i, 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....3...6..11..18..27...
%e 4....2...8..13..20..29...
%e 7....9...5..15..22..31...
%e 12..14..16..10..24..33...
%e 19..21..23..25..17..35...
%e 28..30..32..34..36..26...
%e . . .
%e The start of the sequence as triangle array read by rows:
%e 1;
%e 3,4;
%e 6,2,7;
%e 11,8,9,12;
%e 18,13,5,14,19;
%e 27,20,15,16,21,28;
%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-1)**2+1
%o if i > j:
%o result=(i-1)**2+2*j+1
%o if i < j:
%o result=(j-1)**2+2*i
%Y Cf. A060734, A060736, A185725, A213921, A213922; table T(n,k) contains: in rows A059100, A087475, A114949, A189833, A114948, A114962; in columns A117950, A117951, A117619, A189834, A189836; the main diagonal is A002522.
%K nonn,tabl
%O 1,2
%A _Boris Putievskiy_, Mar 11 2013
|
