

A082233


Square array T(n,k) = 2*n + k, read by antidiagonals in a zigzag fashion, n >= 0 and k >= 1.


3



1, 2, 3, 5, 4, 3, 4, 5, 6, 7, 9, 8, 7, 6, 5, 6, 7, 8, 9, 10, 11, 13, 12, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 15, 14, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11
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OFFSET

0,2


COMMENTS

The nth row contains natural numbers starting from 2n+1. The 2nth column contains even numbers beginning with 2n. The (2n1)th column contains odd numbers beginning with 2n1. The nth antidiagonal sum is given by pentagonal number A000326(n+1). The main diagonal is given by A016777.
For n >= 0 and k >= 1, the term T(n,k) occupies position m = (n+k)*(n+k1)/2 + k*(1  (1)^(n+k))/2 + (n+1)*(1 + (1)^(n+k))/2  1 in the sequence (a(s): s >= 0), i.e., a(m) = T(n,k).  Petros Hadjicostas, Feb 26 2021


LINKS

Ivan Neretin, Table of n, a(n) for n = 0..5049


EXAMPLE

In the following square array (T(n,k): n >= 0, k >= 1), numbers are entered like this: T(0,1), T(0,2), T(1,1), T(2,1), T(1,2), T(0,3), T(0,4), T(1,3), T(2,2), T(3,1), T(4,1), T(3,2), ..., such that every entry is the arithmetic mean of the two diametrically opposite neighbors (wherever such a pair exists).
1 2 3 4 5 6 7 ...
3 4 5 6 7 8 9 ...
5 6 7 8 9 10 11 ...
7 8 9 10 11 12 13 ...
9 10 11 12 13 14 15 ...
...
The sequence (a(n): n >= 0) contains the numbers in the order in which they are entered in the above square array T.


MATHEMATICA

Flatten@Table[If[EvenQ[n], #, Reverse[#]] &[Range[n, 2 n  1]], {n, 11}] (* Ivan Neretin, Aug 24 2017 *)


CROSSREFS

Cf. A000326, A016777, A082234, A229035 (partial sums).
Sequence in context: A330080 A068508 A137403 * A330806 A058981 A117339
Adjacent sequences: A082230 A082231 A082232 * A082234 A082235 A082236


KEYWORD

easy,nonn,tabl


AUTHOR

Amarnath Murthy, Apr 10 2003


EXTENSIONS

More terms from Michel Marcus, Jan 20 2013
New definition from Joerg Arndt and Michel Marcus, Jan 20 2013, corrected R. J. Mathar, Sep 21 2013


STATUS

approved



