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A139600
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Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
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49
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0, 0, 1, 0, 1, 2, 0, 1, 3, 3, 0, 1, 4, 6, 4, 0, 1, 5, 9, 10, 5, 0, 1, 6, 12, 16, 15, 6, 0, 1, 7, 15, 22, 25, 21, 7, 0, 1, 8, 18, 28, 35, 36, 28, 8, 0, 1, 9, 21, 34, 45, 51, 49, 36, 9, 0, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 0, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11
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OFFSET
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0,6
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COMMENTS
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A general formula for polygonal numbers is P(n,k) = (n-2)*(k-1)*k/2 + k, where P(n,k) is the k-th n-gonal number.
The triangle sums, see A180662 for their definitions, link this square array read by antidiagonals with twelve different sequences, see the crossrefs. Most triangle sums are linear sums of shifted combinations of a sequence, see e.g. A189374. - Johannes W. Meijer, Apr 29 2011
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LINKS
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FORMULA
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T(n,k) = n*(k-1)*k/2+k.
G.f.: y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3).
E.g.f.: exp(x+y)*y*(2 + x*y)/2. (End)
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EXAMPLE
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The square array of nonnegatives together with polygonal numbers begins:
=========================================================
....................... A A . . A A A A
....................... 0 0 . . 0 0 1 1
....................... 0 0 . . 1 1 3 3
....................... 0 0 . . 6 7 9 9
....................... 0 0 . . 9 3 6 6
....................... 0 1 . . 5 2 0 0
....................... 4 2 . . 7 9 6 7
=========================================================
Nonnegatives . A001477: 0, 1, 2, 3, 4, 5, 6, 7, ...
Triangulars .. A000217: 0, 1, 3, 6, 10, 15, 21, 28, ...
Squares ...... A000290: 0, 1, 4, 9, 16, 25, 36, 49, ...
Pentagonals .. A000326: 0, 1, 5, 12, 22, 35, 51, 70, ...
Hexagonals ... A000384: 0, 1, 6, 15, 28, 45, 66, 91, ...
Heptagonals .. A000566: 0, 1, 7, 18, 34, 55, 81, 112, ...
Octagonals ... A000567: 0, 1, 8, 21, 40, 65, 96, 133, ...
9-gonals ..... A001106: 0, 1, 9, 24, 46, 75, 111, 154, ...
10-gonals .... A001107: 0, 1, 10, 27, 52, 85, 126, 175, ...
11-gonals .... A051682: 0, 1, 11, 30, 58, 95, 141, 196, ...
12-gonals .... A051624: 0, 1, 12, 33, 64, 105, 156, 217, ...
...
=========================================================
The column with the numbers 2, 3, 4, 5, 6, ... is formed by the numbers > 1 of A000027. The column with the numbers 3, 6, 9, 12, 15, ... is formed by the positive members of A008585.
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MAPLE
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T:= (n, k)-> n*(k-1)*k/2+k:
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MATHEMATICA
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T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[T[n - k - 1, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
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PROG
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(Python)
def A139600Row(n):
x, y = 1, 1
yield 0
while True:
yield x
x, y = x + y + n, y + n
for n in range(8):
R = A139600Row(n)
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CROSSREFS
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A formal extension negative n is in A326728.
Triangle sums (see the comments): A055795 (Row1), A080956 (Row2; terms doubled), A096338 (Kn11, Kn12, Kn13, Fi1, Ze1), A002624 (Kn21, Kn22, Kn23, Fi2, Ze2), A000332 (Kn3, Ca3, Gi3), A134393 (Kn4), A189374 (Ca1, Ze3), A011779 (Ca2, Ze4), A101357 (Ca4), A189375 (Gi1), A189376 (Gi2), A006484 (Gi4). - Johannes W. Meijer, Apr 29 2011
Sequences of m-gonal numbers: A000217 (m=3), A000290 (m=4), A000326 (m=5), A000384 (m=6), A000566 (m=7), A000567 (m=8), A001106 (m=9), A001107 (m=10), A051682 (m=11), A051624 (m=12), A051865 (m=13), A051866 (m=14), A051867 (m=15), A051868 (m=16), A051869 (m=17), A051870 (m=18), A051871 (m=19), A051872 (m=20), A051873 (m=21), A051874 (m=22), A051875 (m=23), A051876 (m=24), A255184 (m=25), A255185 (m=26), A255186 (m=27), A161935 (m=28), A255187 (m=29), A254474 (m=30).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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