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A139600
Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
51
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
OFFSET
0,6
COMMENTS
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
LINKS
Peter Luschny, Figurate number — a very short introduction. With plots from Stefan Friedrich Birkner.
Omar E. Pol, Polygonal numbers, An alternative illustration of initial terms.
FORMULA
T(n,k) = n*(k-1)*k/2+k.
T(n,k) = A057145(n+2,k). - R. J. Mathar, Jul 28 2016
From Stefano Spezia, Apr 12 2024: (Start)
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)
EXAMPLE
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.
MAPLE
T:= (n, k)-> n*(k-1)*k/2+k:
seq(seq(T(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, Oct 14 2018
MATHEMATICA
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 *)
PROG
(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)
print([next(R) for _ in range(11)]) # Peter Luschny, Aug 04 2019
(Magma)
T:= func< n, k | k*(n*(k-1)+2)/2 >;
A139600:= func< n, k | T(n-k, k) >;
[A139600(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024
(SageMath)
def T(n, k): return k*(n*(k-1)+2)/2
def A139600(n, k): return T(n-k, k)
flatten([[A139600(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024
CROSSREFS
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).
Sequence in context: A391660 A089112 A155584 * A198321 A325003 A166278
KEYWORD
nonn,tabl,easy
AUTHOR
Omar E. Pol, Apr 27 2008
EXTENSIONS
Edited by Omar E. Pol, Jan 05 2009
STATUS
approved