

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
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OFFSET

1,2


COMMENTS

A simple permutation of natural numbers.
Parity of the sequence is given by A057211 (nth run has length n).  Jeremy Gardiner, Dec 26 2008
The square with corners T(1,1)=1 and T(n,n)=n^2n+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

Alois P. Heinz, Rows n = 1..141 of triangle, flattened
Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.
Eric W. Weisstein, MathWorld: Pairing functions
Index entries for sequences that are permutations of the natural numbers


FORMULA

T(n,k) = (n1)^2+k, T(k, n)=n^2+1k, 1 <= k <= n.
From Clark Kimberling, Feb 01 2011: (Start)
T(1,k) = k^2 (A000290).
T(n,n) = n^2n+1 (A002061).
T(n,1) = (n1)^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^2n+1, (n1)^2+k):
seq(seq(T(n, dn), n=1..d1), d=2..15);


MATHEMATICA

f[n_, k_]:=k^2n+1/; k>=n;
f[n_, k_]:=(n1)^2+k/; k<n;
TableForm[Table[f[n, k], {n, 1, 10}, {k, 1, 15}]]
Table[f[nk+1, k], {n, 14}, {k, n, 1, 1}]//Flatten (* Clark Kimberling, Feb 01 2011 *)


CROSSREFS

Cf. A060736. Inverse: A064790.
Cf. A185725, A185726, A185728.
Sequence in context: A077809 A201281 A095303 * A075594 A076022 A064421
Adjacent sequences: A060731 A060732 A060733 * A060735 A060736 A060737


KEYWORD

nonn,tabl


AUTHOR

Frank Ellermann, Apr 23 2001


EXTENSIONS

Corrected by Jeremy Gardiner, Dec 26 2008


STATUS

approved



