cp's OEIS Frontend

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

%I A101321 #38 Mar 23 2020 09:43:02
%S A101321 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 A101321 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 A101321 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 A101321 Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.
%C A101321 Row n gives the centered figurate numbers of the n-gon.
%C A101321 Antidiagonal sums are in A101338.
%F A101321 T(n,2) = A016777(n). T(n,3) = A016921(n). T(n,4) = A017281(n).
%F A101321 T(10,m) = A062786(m+1).
%F A101321 T(11,m) = A069125(m+1).
%F A101321 T(12,m) = A003154(m+1).
%F A101321 T(13,m) = A069126(m+1).
%F A101321 T(14,m) = A069127(m+1).
%F A101321 T(15,m) = A069128(m+1).
%F A101321 T(16,m) = A069129(m+1).
%F A101321 T(17,m) = A069130(m+1).
%F A101321 T(18,m) = A069131(m+1).
%F A101321 T(19,m) = A069132(m+1).
%F A101321 T(20,m) = A069133(m+1).
%F A101321 T(n+1,m) = T(n,m) + m*(m+1)/2. - _Gary W. Adamson_ and _Michel Marcus_, Oct 13 2015
%e A101321 The upper left corner of the infinite array T is
%e A101321 |0| 1   1   1   1   1   1   1   1   1   1 ... A000012
%e A101321 |1| 1   2   4   7  11  16  22  29  37  46 ... A000124
%e A101321 |2| 1   3   7  13  21  31  43  57  73  91 ... A002061
%e A101321 |3| 1   4  10  19  31  46  64  85 109 136 ... A005448
%e A101321 |4| 1   5  13  25  41  61  85 113 145 181 ... A001844
%e A101321 |5| 1   6  16  31  51  76 106 141 181 226 ... A005891
%e A101321 |6| 1   7  19  37  61  91 127 169 217 271 ... A003215
%e A101321 |7| 1   8  22  43  71 106 148 197 253 316 ... A069099
%e A101321 |8| 1   9  25  49  81 121 169 225 289 361 ... A016754
%e A101321 |9| 1  10  28  55  91 136 190 253 325 406 ... A060544
%p A101321 A101321 := proc(n,k)
%p A101321         n*k*(k+1)/2+1 ;
%p A101321 end proc: # _R. J. Mathar_, Jul 28 2016
%t A101321 T[n_, m_] := 1 + n m (m + 1)/2;
%t A101321 Table[T[n - m, m], {n, 0, 12}, {m, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 23 2020 *)
%o A101321 (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 A101321 (PARI) T(n,m) = 1 + n*m*(m+1)/2 \\ _Charles R Greathouse IV_, Jul 28 2016
%Y A101321 Cf. A000217, A001263, A101338.
%K A101321 easy,nonn,tabl
%O A101321 0,5
%A A101321 Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004
%E A101321 Edited by _R. J. Mathar_, Oct 21 2009