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A303301
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Square array T(n,k) read by antidiagonals upwards in which row n is obtained by taking the general formula for generalized n-gonal numbers: m*((n - 2)*m - n + 4)/2, where m = 0, +1, -1, +2, -2, +3, -3, ... and n >= 5. Here n >= 0.
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5
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0, 0, 1, 0, 1, -3, 0, 1, -2, 0, 0, 1, -1, 1, -8, 0, 1, 0, 2, -5, -3, 0, 1, 1, 3, -2, 0, -15, 0, 1, 2, 4, 1, 3, -9, -8, 0, 1, 3, 5, 4, 6, -3, -2, -24, 0, 1, 4, 6, 7, 9, 3, 4, -14, -15, 0, 1, 5, 7, 10, 12, 9, 10, -4, -5, -35, 0, 1, 6, 8, 13, 15, 15, 16, 6, 5, -20, -24, 0, 1, 7, 9, 16, 18, 21, 22, 16, 15, -5, -9, -48
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OFFSET
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0,6
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COMMENTS
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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).
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LINKS
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FORMULA
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EXAMPLE
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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...
...
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MATHEMATICA
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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 *)
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CROSSREFS
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Column 3 gives A001477 which coincides with the row numbers.
Row 3 gives 0 together with A008795.
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).
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KEYWORD
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AUTHOR
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STATUS
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approved
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