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 A057145 Square array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, read by antidiagonals upwards. 63
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,3 COMMENTS The set of the "nontrivial" entries T(n>=3,k>=3) is in A090466. - R. J. Mathar, Jul 28 2016 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 REFERENCES 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. LINKS T. D. Noe, Rows n = 2..100, flattened Lukas Andritsch, Boundary algebra of a GL_m-dimer, arXiv:1804.07243 [math.RT], 2018. FORMULA 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 From Wolfdieter Lang, Nov 05 2014: (Start) 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) T(n,k) = A139600(n-2,k) = A086270(n-2,k). - R. J. Mathar, Jul 28 2016 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 EXAMPLE 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, ... ------------------------------------------------------- From Wolfdieter Lang, Nov 04 2014: (Start) 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) . From Omar E. Pol, May 03 2020: (Start) 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) MAPLE A057145 := proc(n, k)     ((n-2)*k^2-(n-4)*k)/2 ; end proc: seq(seq(A057145(d-k, k), k=1..d-2), d=3..12); # R. J. Mathar, Jul 28 2016 MATHEMATICA nn = 12; Flatten[Table[k (3 - k^2 - n + k*n)/2, {n, 2, nn}, {k, n - 1}]] (* T. D. Noe, Oct 10 2012 *) PROG (Magma) /* As square array: */ t:=func; [[t(s, n): s in [1..11]]: n in [2..14]]; // Bruno Berselli, Jun 24 2013 CROSSREFS Many rows and columns of this array are in the database. Cf. A055795 (antidiagonal sums), A064808 (main diagonal). Sequence in context: A131251 A144400 A225281 * A134394 A322967 A284855 Adjacent sequences:  A057142 A057143 A057144 * A057146 A057147 A057148 KEYWORD nonn,nice,tabl,easy AUTHOR N. J. A. Sloane, Sep 12 2000 EXTENSIONS a(50)=49 corrected to a(50)=40 by Jean-François Alcover, Jul 22 2011 Edited: Name shortened, offset in Paul Barry's g.f. corrected and Conway-Guy reference added. - Wolfdieter Lang, Nov 04 2014 STATUS approved

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Last modified May 20 11:16 EDT 2022. Contains 353871 sequences. (Running on oeis4.)