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

0,5

Comments

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

Antidiagonal sums are in A101338.

Links

Table of n, a(n) for n=0..79.

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

Example

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

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 *)

Prog

(Iverson's J language) Let cfn be the formula above. Then the first 20 rows and columns of T are: T =: cfn / ~ i. 20 where i.
(PARI) T(n, m) = 1 + n*m*(m+1)/2 \\ Charles R Greathouse IV, Jul 28 2016

Crossrefs

Cf. A000217, A001263, A101338.

Sequence in context: A058717 A034371 A318951 * A210764 A091186 A138155

Adjacent sequences: A101318 A101319 A101320 * A101322 A101323 A101324

Keyword

easy,nonn,tabl

Author

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

Extensions

Edited by R. J. Mathar, Oct 21 2009

Status

approved