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A057145
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Square array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, read by antidiagonals upwards.
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68
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1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 9, 10, 5, 1, 6, 12, 16, 15, 6, 1, 7, 15, 22, 25, 21, 7, 1, 8, 18, 28, 35, 36, 28, 8, 1, 9, 21, 34, 45, 51, 49, 36, 9, 1, 10, 24, 40, 55, 66, 70, 64, 45, 10, 1, 11, 27, 46, 65, 81, 91, 92, 81, 55, 11, 1, 12, 30, 52, 75, 96, 112
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OFFSET
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2,3
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COMMENTS
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T(n,k) is the smallest number that can be expressed as the sum of k consecutive positive integers that differ by n - 2. In other words: T(n,k) is the sum of k terms of the arithmetic progression with common difference n - 2 and 1st term 1, (see the example). - Omar E. Pol, Apr 29 2020
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REFERENCES
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A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 189, 1966.
J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag (Copernicus), p. 38, 1996.
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LINKS
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FORMULA
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T(2n+4,n) = n^3. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Aug 28 2000
T(n, k) = T(n-1, k) + k*(k-1)/2 [with T(2, k)=k] = T(n, k-1) + 1 + (n-2)*(k-1) [with T(n, 0)=0] = k + (n-2)k(k-1)/2 = k + A063212(n-2, k-1). - Henry Bottomley, Jul 11 2001
G.f. for row n: x*(1+(n-3)*x)/(1-x)^3, n>=2. - Paul Barry, Feb 21 2003
The triangle is a(n, m) = T(n-m+1, m) = (1/2)*m*(n*(m-1) + 3 - m^2) for n >= 2, m = 1, 2, ..., n-1 and zero elsewhere.
O.g.f. for column m (without leading zeros): (x*binomial(m,2) + (1+2*m-m^2)*(m/2)*(1-x))/(x^(m-1)*(1-x)^2). (End)
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EXAMPLE
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Array T(n k) (n >= 2, k >= 1) begins:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, ...
1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, ...
1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, ...
1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, ...
1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, ...
1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, ...
1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, ...
1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, ...
1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, ...
1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, ...
-------------------------------------------------------
The triangle a(k, m) begins:
k\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
2: 1
3: 1 2
4: 1 3 3
5: 1 4 6 4
6: 1 5 9 10 5
7: 1 6 12 16 15 6
8: 1 7 15 22 25 21 7
9: 1 8 18 28 35 36 28 8
10: 1 9 21 34 45 51 49 36 9
11: 1 10 24 40 55 66 70 64 45 10
12: 1 11 27 46 65 81 91 92 81 55 11
13: 1 12 30 52 75 96 112 120 117 100 66 12
14: 1 13 33 58 85 111 133 148 153 145 121 78 13
15: 1 14 36 64 95 126 154 176 189 190 176 144 91 14
...
-------------------------------------------------------
a(2,1) = T(2,1), a(6, 3) = T(4, 3). (End)
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Illustration of the corner of the square array:
.
1 2 3 4
O O O O O O O O O O
.
1 3 6 10
O O O O O O O O O O
O O O O O O
O O O
O
.
1 4 9 16
O O O O O O O O O O
O O O O O O
O O O O O O
O O O
O O O
O
O
.
1 5 12 22
O O O O O O O O O O
O O O O O O
O O O O O O
O O O O O O
O O O
O O O
O O O
O
O
O
(End)
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MAPLE
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((n-2)*k^2-(n-4)*k)/2 ;
end proc:
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MATHEMATICA
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nn = 12; Flatten[Table[k (3 - k^2 - n + k*n)/2, {n, 2, nn}, {k, n - 1}]] (* T. D. Noe, Oct 10 2012 *)
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PROG
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(Magma) /* As square array: */ t:=func<n, s | (n^2*(s-2)-n*(s-4))/2>; [[t(s, n): s in [1..11]]: n in [2..14]]; // Bruno Berselli, Jun 24 2013
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CROSSREFS
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Many rows and columns of this array are in the database.
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KEYWORD
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AUTHOR
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EXTENSIONS
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Edited: Name shortened, offset in Paul Barry's g.f. corrected and Conway-Guy reference added. - Wolfdieter Lang, Nov 04 2014
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STATUS
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approved
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