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. - Omar E. Pol, Dec 21 2008
LINKS
G. C. Greubel, Table of n, a(n) for the first 50 rows, 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.
FORMULA
T(n,k) = A086270(n,k), k>0. - R. J. Mathar, Aug 06 2008
T(n,k) = (n+1)*(k-1)*k/2 +k, n>=0, k>=0. - Omar E. Pol, Jan 07 2009
From G. C. Greubel, Jul 12 2024: (Start)
t(n, k) = (k/2)*( (k-1)*(n-k+1) + 2), where t(n,k) is this array read by rising antidiagonals.
t(2*n, n) = A006003(n).
t(2*n+1, n) = A002411(n).
t(2*n-1, n) = A006000(n-1).
Sum_{k=0..n} t(n, k) = A006522(n+2).
Sum_{k=0..n} (-1)^k*t(n, k) = (-1)^n * A117142(n).
Sum_{k=0..n} t(n-k, k) = (2*n^4 + 34*n^2 + 48*n - 15 + 3*(-1)^n*(2*n^2 + 16*n + 5))/384. (End)
EXAMPLE
The square array of polygonal numbers begins:
========================================================
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,
And so on ..............................................
========================================================
MATHEMATICA
T[n_, k_] := (n + 1)*(k - 1)*k/2 + k; Table[ T[n - k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Robert G. Wilson v, Jul 12 2009 *)
PROG
(Magma)
T:= func< n, k | k*((n+1)*(k-1) +2)/2 >;
A139601:= func< n, k | T(n-k, k) >;
[A139601(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 12 2024
(SageMath)
def T(n, k): return k*((n+1)*(k-1)+2)/2
def A139601(n, k): return T(n-k, k)
flatten([[A139601(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jul 12 2024
CROSSREFS
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).
KEYWORD
AUTHOR
Omar E. Pol, Apr 27 2008
STATUS
approved