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A057145 Array of polygonal numbers T(n,k) = ((n-2)*k^2 - (n-4)*k)/2, n >= 2, k >= 1, read by antidiagonals. 58
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

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.

Index to sequences related to polygonal numbers

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)

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<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

CROSSREFS

Many rows and columns of this array are in the database.

Cf. A055795 (antidiagonal sums).

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-Fran├žois 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 October 18 15:41 EDT 2019. Contains 328162 sequences. (Running on oeis4.)