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%I #110 Apr 16 2024 14:02:02
%S 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,
%T 18,28,35,36,28,8,1,9,21,34,45,51,49,36,9,1,10,24,40,55,66,70,64,45,
%U 10,1,11,27,46,65,81,91,92,81,55,11,1,12,30,52,75,96,112
%N Square array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, read by antidiagonals upwards.
%C The set of the "nontrivial" entries T(n>=3,k>=3) is in A090466. - _R. J. Mathar_, Jul 28 2016
%C 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
%D A. H. Beiler, Recreations in the Theory of Numbers. New York: Dover, p. 189, 1966.
%D J. H. Conway and R. K. Guy, The Book of Numbers, Springer-Verlag (Copernicus), p. 38, 1996.
%H T. D. Noe, <a href="/A057145/b057145.txt">Rows n = 2..100, flattened</a>
%H Lukas Andritsch, <a href="https://arxiv.org/abs/1804.07243">Boundary algebra of a GL_m-dimer</a>, arXiv:1804.07243 [math.RT], 2018.
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F T(2n+4,n) = n^3. - Stuart M. Ellerstein (ellerstein(AT)aol.com), Aug 28 2000
%F 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
%F G.f. for row n: x*(1+(n-3)*x)/(1-x)^3, n>=2. - _Paul Barry_, Feb 21 2003
%F From _Wolfdieter Lang_, Nov 05 2014: (Start)
%F 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.
%F 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)
%F T(n,k) = A139600(n-2,k) = A086270(n-2,k). - _R. J. Mathar_, Jul 28 2016
%F Row sums of A077028: T(n+2,k+1) = Sum_{j=0..k} A077028(n,j), where A077028(n,k) = 1+n*k is the square array interpretation of A077028 (the 1D polygonal numbers). - _R. J. Mathar_, Jul 30 2016
%F G.f.: x^2*y*(1 - x - y + 2*x*y)/((1 - x)^2*(1 - y)^3). - _Stefano Spezia_, Apr 12 2024
%e Array T(n k) (n >= 2, k >= 1) begins:
%e 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...
%e 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, ...
%e 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, ...
%e 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, ...
%e 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, ...
%e 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, ...
%e 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, ...
%e 1, 9, 24, 46, 75, 111, 154, 204, 261, 325, 396, ...
%e 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, ...
%e 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, ...
%e 1, 12, 33, 64, 105, 156, 217, 288, 369, 460, 561, ...
%e 1, 13, 36, 70, 115, 171, 238, 316, 405, 505, 616, ...
%e 1, 14, 39, 76, 125, 186, 259, 344, 441, 550, 671, ...
%e -------------------------------------------------------
%e From _Wolfdieter Lang_, Nov 04 2014: (Start)
%e The triangle a(k, m) begins:
%e k\m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ...
%e 2: 1
%e 3: 1 2
%e 4: 1 3 3
%e 5: 1 4 6 4
%e 6: 1 5 9 10 5
%e 7: 1 6 12 16 15 6
%e 8: 1 7 15 22 25 21 7
%e 9: 1 8 18 28 35 36 28 8
%e 10: 1 9 21 34 45 51 49 36 9
%e 11: 1 10 24 40 55 66 70 64 45 10
%e 12: 1 11 27 46 65 81 91 92 81 55 11
%e 13: 1 12 30 52 75 96 112 120 117 100 66 12
%e 14: 1 13 33 58 85 111 133 148 153 145 121 78 13
%e 15: 1 14 36 64 95 126 154 176 189 190 176 144 91 14
%e ...
%e -------------------------------------------------------
%e a(2,1) = T(2,1), a(6, 3) = T(4, 3). (End)
%e .
%e From _Omar E. Pol_, May 03 2020: (Start)
%e Illustration of the corner of the square array:
%e .
%e 1 2 3 4
%e O O O O O O O O O O
%e .
%e 1 3 6 10
%e O O O O O O O O O O
%e O O O O O O
%e O O O
%e O
%e .
%e 1 4 9 16
%e O O O O O O O O O O
%e O O O O O O
%e O O O O O O
%e O O O
%e O O O
%e O
%e O
%e .
%e 1 5 12 22
%e O O O O O O O O O O
%e O O O O O O
%e O O O O O O
%e O O O O O O
%e O O O
%e O O O
%e O O O
%e O
%e O
%e O
%e (End)
%p A057145 := proc(n,k)
%p ((n-2)*k^2-(n-4)*k)/2 ;
%p end proc:
%p seq(seq(A057145(d-k,k),k=1..d-2),d=3..12); # _R. J. Mathar_, Jul 28 2016
%t nn = 12; Flatten[Table[k (3 - k^2 - n + k*n)/2, {n, 2, nn}, {k, n - 1}]] (* _T. D. Noe_, Oct 10 2012 *)
%o (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
%Y Many rows and columns of this array are in the database.
%Y Cf. A055795 (antidiagonal sums), A064808 (main diagonal).
%Y Cf. A063212, A077028, A086270, A090466, A139600.
%K nonn,nice,tabl,easy,changed
%O 2,3
%A _N. J. A. Sloane_, Sep 12 2000
%E a(50)=49 corrected to a(50)=40 by _Jean-François Alcover_, Jul 22 2011
%E 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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