A215631 Triangle read by rows: T(n,k) = n^2 + n*k + k^2, 1 <= k <= n.

3, 7, 12, 13, 19, 27, 21, 28, 37, 48, 31, 39, 49, 61, 75, 43, 52, 63, 76, 91, 108, 57, 67, 79, 93, 109, 127, 147, 73, 84, 97, 112, 129, 148, 169, 192, 91, 103, 117, 133, 151, 171, 193, 217, 243, 111, 124, 139, 156, 175, 196, 219, 244, 271, 300, 133, 147, 163

Offset: 1

Author

Reinhard Zumkeller, Nov 11 2012

Keywords

• nonn
• tabl
• easy

Examples

			The triangle begins:
row n   T(n,k), 1 <= k <= n
   1:     3
   2:     7   12
   3:    13   19   27
   4:    21   28   37   48
   5:    31   39   49   61   75
   6:    43   52   63   76   91  108
   7:    57   67   79   93  109  127  147
   8:    73   84   97  112  129  148  169  192
   9:    91  103  117  133  151  171  193  217  243
  10:   111  124  139  156  175  196  219  244  271  300
  11:   133  147  163  181  201  223  247  273  301  331  363
  12:   157  172  189  208  229  252  277  304  333  364  397  432
		

Links

• Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Crossrefs

Cf. A215646 (row sums), A002061 (left edge, shifted), A033428 (right edge), A003215.

Programs

Haskell

a215631 n k = a215631_tabl !! (n-1) !! (k-1)
a215631_row n = a215631_tabl !! (n-1)
a215631_tabl = zipWith3 (zipWith3 (\u v w -> u + v + w))
                        a093995_tabl a075362_tabl a133819_tabl
-- Reinhard Zumkeller, Nov 11 2012

Magma

[[i^2+i*j+j^2: j in [1..i]]: i in [1..10]]; // Vincenzo Librandi, Jun 07 2015

Maple

seq(seq(i^2+i*j+j^2, j=1..i),i=1..10); # Robert Israel, May 10 2015

Mathematica

Table[n^2 + n*k + k^2, {n, 11}, {k, n}] // Flatten (* Michael De Vlieger, May 12 2015 *)

PARI

for(n=1,15,for(k=1,n,print1(n^2+n*k+k^2,", "))) \\ Derek Orr, May 13 2015

Formula

T(n,k) = 2*A070216(n,k) - A215630(n,k).

G.f. for triangle: (3-2*x+3*x*y+x^2-11*x^2*y+4*x^3*y+x^3*y^2+x^4*y^2)*x*y/((1-x)^3*(1-x*y)^3). - Robert Israel, May 10 2015

From Avi Friedlich, May 26 2015: (Start)

T(n,k) = A093995(n,k) + A075362(n,k) + A133819(n,k).

T(k+1,k) = A003215(k).

T(k+2,k) = A003215(k)/2 + A003215(k+1)/2.

T(k+3,k) = A003215(k)/3 + A003215(k+1)/3 + A003215(k+2)/3 and so on. (End)