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A060734 Natural numbers written as a square array ending in last row from left to right and rightmost column from bottom to top are read by antidiagonals downwards. 15
1, 4, 2, 9, 3, 5, 16, 8, 6, 10, 25, 15, 7, 11, 17, 36, 24, 14, 12, 18, 26, 49, 35, 23, 13, 19, 27, 37, 64, 48, 34, 22, 20, 28, 38, 50, 81, 63, 47, 33, 21, 29, 39, 51, 65, 100, 80, 62, 46, 32, 30, 40, 52, 66, 82, 121, 99, 79, 61, 45, 31, 41, 53, 67, 83, 101 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
A simple permutation of natural numbers.
Parity of the sequence is given by A057211 (n-th run has length n). - Jeremy Gardiner, Dec 26 2008
The square with corners T(1,1)=1 and T(n,n)=n^2-n+1 is occupied by the numbers 1,2,...,n^2. - Clark Kimberling, Feb 01 2011
a(n) is pairing function - function that reversibly maps Z^{+} x Z^{+} onto Z^{+}, where Z^{+} - the set of integer positive numbers. - Boris Putievskiy, Dec 17 2012
LINKS
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
FORMULA
T(n,k) = (n-1)^2+k, T(k, n)=n^2+1-k, 1 <= k <= n.
From Clark Kimberling, Feb 01 2011: (Start)
T(1,k) = k^2 (A000290).
T(n,n) = n^2-n+1 (A002061).
T(n,1) = (n-1)^2+1 (A002522). (End)
EXAMPLE
Northwest corner:
.1 4 9 16 .. => a(1) = 1
.2 3 8 15 .. => a(2) = 4, a(3) = 2
.5 6 7 14 .. => a(4) = 9, a(5) = 3, a(6) = 5
10 11 12 13 .. => a(7) = 16, a(8) = 8, a(9) = 6, a(10)=10
MAPLE
T:= (n, k)-> `if`(n<=k, k^2-n+1, (n-1)^2+k):
seq(seq(T(n, d-n), n=1..d-1), d=2..15);
MATHEMATICA
f[n_, k_]:=k^2-n+1/; k>=n;
f[n_, k_]:=(n-1)^2+k/; k<n;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[n-k+1, k], {n, 14}, {k, n, 1, -1}]//Flatten (* Clark Kimberling, Feb 01 2011 *)
CROSSREFS
Cf. A060736. Inverse: A064790.
Sequence in context: A077809 A201281 A095303 * A075594 A076022 A064421
KEYWORD
nonn,tabl
AUTHOR
Frank Ellermann, Apr 23 2001
EXTENSIONS
Corrected by Jeremy Gardiner, Dec 26 2008
STATUS
approved

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Last modified March 29 09:32 EDT 2024. Contains 371268 sequences. (Running on oeis4.)