A101321 - OEIS

%I #38 Mar 23 2020 09:43:02

%S 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,

%T 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,

%U 46,73,85,85,76,61,43,25,10,1,1,56,91,109,113,106,91,71,49,28,11,1,1,67

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

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

%C Antidiagonal sums are in A101338.

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

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

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

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

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

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

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

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

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

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

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

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

%F T(n+1,m) = T(n,m) + m*(m+1)/2. - _Gary W. Adamson_ and _Michel Marcus_, Oct 13 2015

%e The upper left corner of the infinite array T is

%e |0| 1   1   1   1   1   1   1   1   1   1 ... A000012

%e |1| 1   2   4   7  11  16  22  29  37  46 ... A000124

%e |2| 1   3   7  13  21  31  43  57  73  91 ... A002061

%e |3| 1   4  10  19  31  46  64  85 109 136 ... A005448

%e |4| 1   5  13  25  41  61  85 113 145 181 ... A001844

%e |5| 1   6  16  31  51  76 106 141 181 226 ... A005891

%e |6| 1   7  19  37  61  91 127 169 217 271 ... A003215

%e |7| 1   8  22  43  71 106 148 197 253 316 ... A069099

%e |8| 1   9  25  49  81 121 169 225 289 361 ... A016754

%e |9| 1  10  28  55  91 136 190 253 325 406 ... A060544

%p A101321 := proc(n,k)

%p         n*k*(k+1)/2+1 ;

%p end proc: # _R. J. Mathar_, Jul 28 2016

%t T[n_, m_] := 1 + n m (m + 1)/2;

%t Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 23 2020 *)

%o (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.

%o (PARI) T(n,m) = 1 + n*m*(m+1)/2 \\ _Charles R Greathouse IV_, Jul 28 2016

%Y Cf. A000217, A001263, A101338.

%K easy,nonn,tabl

%O 0,5

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

%E Edited by _R. J. Mathar_, Oct 21 2009