OFFSET
0,6
COMMENTS
Note that the formula mentioned in the definition gives several kinds of numbers, for example:
Row 2 gives A001057 (canonical enumeration of integers).
Row 3 gives 0 together with A008795 (Molien series for 3-dimensional representation of dihedral group D_6 of order 6).
Row 4 gives A008794 (squares repeated) except the initial zero.
Finally, for n >= 5 row n gives the generalized k-gonal numbers (see Crossrefs section).
LINKS
Alois P. Heinz, Antidiagonals n = 0..200 (first 45 antidiagonals from Robert G. Wilson v)
FORMULA
T(n,k) = A194801(n-3,k) if n >= 3.
EXAMPLE
Array begins:
------------------------------------------------------------------
n\m Seq. No. 0 1 -1 2 -2 3 -3 4 -4 5 -5
------------------------------------------------------------------
0 A317300: 0, 1, -3, 0, -8, -3, -15, -8, -24, -15, -35...
1 A317301: 0, 1, -2, 1, -5, 0, -9, -2, -14, -5, -20...
2 A001057: 0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5...
3 (A008795): 0, 1, 0, 3, 1, 6, 3, 10, 6, 15, 10...
4 (A008794): 0, 1, 1, 4, 4, 9, 9, 16, 16, 25, 25...
5 A001318: 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40...
6 A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55...
7 A085787: 0, 1, 4, 7, 13, 18, 27, 34, 46, 55, 70...
8 A001082: 0, 1, 5, 8, 16, 21, 33, 40, 56, 65, 85...
9 A118277: 0, 1, 6, 9, 19, 24, 39, 46, 66, 75, 100...
10 A074377: 0, 1, 7, 10, 22, 27, 45, 52, 76, 85, 115...
11 A195160: 0, 1, 8, 11, 25, 30, 51, 58, 86, 95, 130...
12 A195162: 0, 1, 9, 12, 28, 33, 57, 64, 96, 105, 145...
13 A195313: 0, 1, 10, 13, 31, 36, 63, 70, 106, 115, 160...
14 A195818: 0, 1, 11, 14, 34, 39, 69, 76, 116, 125, 175...
15 A277082: 0, 1, 12, 15, 37, 42, 75, 82, 126, 135, 190...
...
MATHEMATICA
t[n_, r_] := PolygonalNumber[n, If[OddQ@ r, Floor[(r + 1)/2], -r/2]]; Table[ t[n - r, r], {n, 0, 11}, {r, 0, n}] // Flatten (* also *)
(* to view the square array *) Table[ t[n, r], {n, 0, 15}, {r, 0, 10}] // TableForm (* Robert G. Wilson v, Aug 08 2018 *)
CROSSREFS
Column 3 gives A001477 which coincides with the row numbers.
Main diagonal gives A292551.
Row 3 gives 0 together with A008795.
Row 4 gives A008794.
For n >= 5, rows n gives the generalized n-gonal numbers: A001318 (n=5), A000217 (n=6), A085787 (n=7), A001082 (n=8), A118277 (n=9), A074377 (n=10), A195160 (n=11), A195162 (n=12), A195313 (n=13), A195818 (n=14), A277082 (n=15), A274978 (n=16), A303305 (n=17), A274979 (n=18), A303813 (n=19), A218864 (n=20), A303298 (n=21), A303299 (n=22), A303303 (n=23), A303814 (n=24), A303304 (n=25), A316724 (n=26), A316725 (n=27), A303812 (n=28), A303815 (n=29), A316729 (n=30).
Cf. A317302 (a similar table but with polygonal numbers).
KEYWORD
sign,tabl
AUTHOR
Omar E. Pol, Jun 08 2018
STATUS
approved