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A101321
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Table T(n,m) = 1 + n*m*(m+1)/2 read by antidiagonals: centered polygonal numbers.
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18
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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
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OFFSET
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0,5
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COMMENTS
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Row n gives the centered figurate numbers of the n-gon.
Antidiagonal sums are in A101338.
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LINKS
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Table of n, a(n) for n=0..79.
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FORMULA
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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
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EXAMPLE
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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
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MAPLE
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A101321 := proc(n, k)
n*k*(k+1)/2+1 ;
end proc: # R. J. Mathar, Jul 28 2016
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A000217, A001263, A101338.
Sequence in context: A034367 A034371 A318951 * A210764 A091186 A138155
Adjacent sequences: A101318 A101319 A101320 * A101322 A101323 A101324
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Eugene McDonnell (eemcd(AT)mac.com), Dec 24 2004
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EXTENSIONS
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Edited by R. J. Mathar, Oct 21 2009
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STATUS
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approved
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