Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #42 Dec 03 2018 18:28:00
%S 1,3,5,2,4,6,8,10,12,14,7,9,11,13,15,17,19,21,23,25,27,16,18,20,22,24,
%T 26,28,30,32,34,36,38,40,42,44,29,31,33,35,37,39,41,43,45,47,49,51,53,
%U 55,57,59,61,63,65,46,48,50,52,54,56,58,60,62,64,66,68
%N Array read by downward antidiagonals: T(n,k) = (n+k-1)*(n+k-2) + n + floor((n+k)/2)*(1-2*floor((n+k)/2)), for n, k > 0
%C Enumeration table T(n,k). The order of the list: T(1,1)=1; for k>0: T(1,2*k+1),T(1,2*k); T(2,2*k),T(2,2*k-1); ... T(2*k,2),T(2*k,1); T(2*k+1,1).
%C The order of the list is descent stairs from the northeast to southwest: step to the west, step to the south, step to the west and so on. The length of each step is 1 or alternation of elements pair adjacent 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.
%H Boris Putievskiy, <a href="/A210521/b210521.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 W. Weisstein, <a href="http://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: T(n,k) = (n+k-1)*(n+k-2) + 2*n + floor((n+k)/2)*(1-2*floor((n+k)/2)).
%F As a linear sequence: a(n) = 2*A000027(n) + A204164(n)*(1-2*A204164(n)).
%F a(n) = 2*n+v*(1-2*v), where t = floor((-1+sqrt(8*n-7))/2) and v = floor((t+2)/2).
%F G.f. as a table: (2 - 2*y - 5*y^2 + 6*y^3 + 3*y^4 + x*y*(1 + 3*y-5*y^2 + y^3) + x^2*(- 3 + 7*y + 5*y^2 - 11*y^3 - 6*y^4) - x^3*(- 4 + 5*y + 7*y^2 - 9*y^3 + y^4) + x^4*(1 - y - 4*y^2 + y^3 + 7*y^4))/((- 1 + x)^3*(1 + x)^2*(- 1 + y)^3*(1 + y)^2). - _Stefano Spezia_, Dec 03 2018
%e The start of the sequence as a table:
%e 1, 3, 2, 8, 7, 17, 16, 30, 29, ...
%e 5, 4, 10, 9, 19, 18, 32, 31, 49, ...
%e 6, 12, 11, 21, 20, 34, 33, 51, 50, ...
%e 14, 13, 23, 22, 36, 35, 53, 52, 74, ...
%e 15, 25, 24, 38, 37, 55, 54, 76, 75, ...
%e 27, 26, 40, 39, 57, 56, 78, 77, 103, ...
%e 28, 42, 41, 59, 58, 80, 79, 105, 104, ...
%e 44, 43, 61, 60, 82, 81, 107, 106, 136, ...
%e 45, 63, 62, 84, 83, 109, 108, 138, 137, ...
%e ...
%e The start of the sequence as a triangular array read by rows:
%e 1;
%e 3, 5;
%e 2, 4, 6;
%e 8, 10, 12, 14;
%e 7, 9, 11, 13, 15;
%e 17, 19, 21, 23, 25, 27;
%e 16, 18, 20, 22, 24, 26, 28;
%e 30, 32, 34, 36, 38, 40, 42, 44;
%e 29, 31, 33, 35, 37, 39, 41, 43, 45;
%e ...
%e The sequence as array read by rows, the length of row r is 4*r-1. First 2*r-1 numbers are from row 2*r-1 of the triangular array above. Last 2*r numbers are from row 2*r of the triangular array. The start of the sequence:
%e 1,3,5;
%e 2,4,6,8,10,12,14;
%e 7,9,11,13,15,17,19,21,23,25,27;
%e 16,18,20,22,24,26,28,30,32,34,36,38,40,42,44;
%e 29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65;
%e ...
%e Row r contains 4*r-1 numbers: 2*r^2-5*r+4, 2*r^2-5*r+6, 2*r^2-5*r+8, ..., r*(2*r+3).
%e Considered as a triangle, the rows have constant parity.
%t T[n_, k_] := (n+k-1)(n+k-2) + 2n + Floor[(n+k)/2](1 - 2 Floor[(n+k)/2]);
%t Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 03 2018 *)
%o (Python)
%o t=int((math.sqrt(8*n-7)-1)/2)
%o v=int((t+2)/2)
%o result=2*n+v*(1-2*v)
%Y Cf. A000027, A204164, the main diagonal is A084849.
%K nonn,tabl
%O 1,2
%A _Boris Putievskiy_, Jan 26 2013