The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A210521 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 5
 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, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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). 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. Permutation of the natural numbers. 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. LINKS Boris Putievskiy, Rows n = 1..140 of triangle, flattened Boris Putievskiy, Transformations [of] Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012. Eric W. Weisstein, MathWorld: Pairing functions FORMULA As a table: T(n,k) = (n+k-1)*(n+k-2) + 2*n + floor((n+k)/2)*(1-2*floor((n+k)/2)). As a linear sequence: a(n) = 2*A000027(n) + A204164(n)*(1-2*A204164(n)). a(n) = 2*n+v*(1-2*v), where t = floor((-1+sqrt(8*n-7))/2) and v = floor((t+2)/2). 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 EXAMPLE The start of the sequence as a table:    1,  3,  2,  8,  7,  17,  16,  30,  29, ...    5,  4, 10,  9, 19,  18,  32,  31,  49, ...    6, 12, 11, 21, 20,  34,  33,  51,  50, ...   14, 13, 23, 22, 36,  35,  53,  52,  74, ...   15, 25, 24, 38, 37,  55,  54,  76,  75, ...   27, 26, 40, 39, 57,  56,  78,  77, 103, ...   28, 42, 41, 59, 58,  80,  79, 105, 104, ...   44, 43, 61, 60, 82,  81, 107, 106, 136, ...   45, 63, 62, 84, 83, 109, 108, 138, 137, ...   ... The start of the sequence as a triangular array read by rows:    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, 26, 28;   30, 32, 34, 36, 38, 40, 42, 44;   29, 31, 33, 35, 37, 39, 41, 43, 45;   ... 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: 1,3,5; 2,4,6,8,10,12,14; 7,9,11,13,15,17,19,21,23,25,27; 16,18,20,22,24,26,28,30,32,34,36,38,40,42,44; 29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,61,63,65; ... 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). Considered as a triangle, the rows have constant parity. MATHEMATICA T[n_, k_] := (n+k-1)(n+k-2) + 2n + Floor[(n+k)/2](1 - 2 Floor[(n+k)/2]); Table[T[n-k+1, k], {n, 1, 12}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Dec 03 2018 *) PROG (Python) t=int((math.sqrt(8*n-7)-1)/2) v=int((t+2)/2) result=2*n+v*(1-2*v) CROSSREFS Cf. A000027, A204164, the main diagonal is A084849. Sequence in context: A309492 A131793 A065186 * A219249 A203553 A081964 Adjacent sequences:  A210518 A210519 A210520 * A210522 A210523 A210524 KEYWORD nonn,tabl AUTHOR Boris Putievskiy, Jan 26 2013 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified June 5 18:48 EDT 2020. Contains 334854 sequences. (Running on oeis4.)