A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 7, 7, 4, 1, 1, 11, 13, 10, 5, 1, 1, 16, 21, 19, 13, 6, 1, 1, 22, 31, 31, 25, 16, 7, 1, 1, 29, 43, 46, 41, 31, 19, 8, 1, 1, 37, 57, 64, 61, 51, 37, 22, 9, 1, 1, 46, 73, 85, 85, 76, 61, 43, 25, 10, 1, 1, 56, 91, 109, 113, 106, 91, 71, 49, 28, 11, 1, 1, 67

Offset: 0

Author

Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004

Keywords

• easy
• nonn
• tabl

Comments

Row n gives the centered figurate numbers of the n-gon.

Antidiagonal sums are in A101338.

Examples

			The upper left corner of the infinite array T is
|0| 1   1   1   1   1   1   1   1   1   1 ... A000012
|1| 1   2   4   7  11  16  22  29  37  46 ... A000124
|2| 1   3   7  13  21  31  43  57  73  91 ... A002061
|3| 1   4  10  19  31  46  64  85 109 136 ... A005448
|4| 1   5  13  25  41  61  85 113 145 181 ... A001844
|5| 1   6  16  31  51  76 106 141 181 226 ... A005891
|6| 1   7  19  37  61  91 127 169 217 271 ... A003215
|7| 1   8  22  43  71 106 148 197 253 316 ... A069099
|8| 1   9  25  49  81 121 169 225 289 361 ... A016754
|9| 1  10  28  55  91 136 190 253 325 406 ... A060544
		

Crossrefs

Cf. A000217, A001263, A101338.

Programs

Maple

A101321 := proc(n,k)
        n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016

Mathematica

T[n_, m_] := 1 + n m (m + 1)/2;
Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* Jean-François Alcover, Mar 23 2020 *)

PARI

T(n,m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

Formula

T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).

T(10,m) = A062786(m+1).

T(11,m) = A069125(m+1).

T(12,m) = A003154(m+1).

T(13,m) = A069126(m+1).

T(14,m) = A069127(m+1).

T(15,m) = A069128(m+1).

T(16,m) = A069129(m+1).

T(17,m) = A069130(m+1).

T(18,m) = A069131(m+1).

T(19,m) = A069132(m+1).

T(20,m) = A069133(m+1).

T(n+1,m) = T(n,m) + m*(m+1)/2. - Gary W. Adamson and Michel Marcus, Oct 13 2015

Extensions

Edited by R. J. Mathar, Oct 21 2009