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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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The n-th row contains natural numbers starting from 2n+1. The 2n-th column contains even numbers beginning with 2n. The (2n-1)-th column contains odd numbers beginning with 2n-1. The n-th 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+k-1)/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

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Last modified August 3 01:53 EDT 2021. Contains 346429 sequences. (Running on oeis4.)