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A317302
Square array T(n,k) = (n - 2)*(k - 1)*k/2 + k, with n >= 0, k >= 0, read by antidiagonals upwards.
5
0, 0, 1, 0, 1, 0, 0, 1, 1, -3, 0, 1, 2, 0, -8, 0, 1, 3, 3, -2, -15, 0, 1, 4, 6, 4, -5, -24, 0, 1, 5, 9, 10, 5, -9, -35, 0, 1, 6, 12, 16, 15, 6, -14, -48, 0, 1, 7, 15, 22, 25, 21, 7, -20, -63, 0, 1, 8, 18, 28, 35, 36, 28, 8, -27, -80, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, -35, -99, 0, 1, 10, 24, 40, 55, 66
OFFSET
0,10
COMMENTS
Note that the formula gives several kinds of numbers, for example:
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the nonnegative numbers.
For n >= 3, row n gives the n-gonal numbers (see Crossrefs section).
LINKS
Omar E. Pol, Polygonal numbers.
University of Surrey, Dept. of Mathematics, Polygonal Numbers - or Numbers as Shapes.
Eric Weisstein's World of Mathematics, Figurate Number.
Eric Weisstein's World of Mathematics, Polygonal Number.
Wikipedia, Polygonal number.
FORMULA
T(n,k) = A139600(n-2,k) if n >= 2.
T(n,k) = A139601(n-3,k) if n >= 3.
EXAMPLE
Array begins:
------------------------------------------------------------------------
n\k Numbers Seq. No. 0 1 2 3 4 5 6 7 8
------------------------------------------------------------------------
0 ............ (A258837): 0, 1, 0, -3, -8, -15, -24, -35, -48, ...
1 ............ (A080956): 0, 1, 1, 0, -2, -5, -9, -14, -20, ...
2 Nonnegatives A001477: 0, 1, 2, 3, 4, 5, 6, 7, 8, ...
3 Triangulars A000217: 0, 1, 3, 6, 10, 15, 21, 28, 36, ...
4 Squares A000290: 0, 1, 4, 9, 16, 25, 36, 49, 64, ...
5 Pentagonals A000326: 0, 1, 5, 12, 22, 35, 51, 70, 92, ...
6 Hexagonals A000384: 0, 1, 6, 15, 28, 45, 66, 91, 120, ...
7 Heptagonals A000566: 0, 1, 7, 18, 34, 55, 81, 112, 148, ...
8 Octagonals A000567: 0, 1, 8, 21, 40, 65, 96, 133, 176, ...
9 9-gonals A001106: 0, 1, 9, 24, 46, 75, 111, 154, 204, ...
10 10-gonals A001107: 0, 1, 10, 27, 52, 85, 126, 175, 232, ...
11 11-gonals A051682: 0, 1, 11, 30, 58, 95, 141, 196, 260, ...
12 12-gonals A051624: 0, 1, 12, 33, 64, 105, 156, 217, 288, ...
13 13-gonals A051865: 0, 1, 13, 36, 70, 115, 171, 238, 316, ...
14 14-gonals A051866: 0, 1, 14, 39, 76, 125, 186, 259, 344, ...
15 15-gonals A051867: 0, 1, 15, 42, 82, 135, 201, 280, 372, ...
...
CROSSREFS
Column 0 gives A000004.
Column 1 gives A000012.
Column 2 gives A001477, which coincides with the row numbers.
Main diagonal gives A060354.
Row 0 gives 0 together with A258837.
Row 1 gives 0 together with A080956.
Row 2 gives A001477, the same as column 2.
For n >= 3, row n gives the n-gonal numbers: A000217 (n=3), A000290 (n=4), A000326 (n=5), A000384 (n=6), A000566 (n=7), A000567 (n=8), A001106 (n=9), A001107 (n=10), A051682 (n=11), A051624 (n=12), A051865 (n=13), A051866 (n=14), A051867 (n=15), A051868 (n=16), A051869 (n=17), A051870 (n=18), A051871 (n=19), A051872 (n=20), A051873 (n=21), A051874 (n=22), A051875 (n=23), A051876 (n=24), A255184 (n=25), A255185 (n=26), A255186 (n=27), A161935 (n=28), A255187 (n=29), A254474 (n=30).
Cf. A303301 (similar table but with generalized polygonal numbers).
Sequence in context: A124027 A097610 A161556 * A242869 A224878 A129555
KEYWORD
sign,tabl,easy
AUTHOR
Omar E. Pol, Aug 09 2018
STATUS
approved