login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 59th year, we have over 358,000 sequences, and we’ve crossed 10,300 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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. 64
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.

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)

.

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<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), 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 3 09:03 EST 2022. Contains 358515 sequences. (Running on oeis4.)