A139600 Square array T(n,k) = n*(k-1)*k/2+k, of nonnegative numbers together with polygonal numbers, read by antidiagonals upwards.
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
Examples
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.
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- 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.
- Index to sequences related to polygonal numbers
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).
Programs
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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
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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
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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 *) -
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
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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
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)
Extensions
Edited by Omar E. Pol, Jan 05 2009
Comments