A215631 - OEIS

%I #65 Oct 31 2024 01:17:50

%S 3,7,12,13,19,27,21,28,37,48,31,39,49,61,75,43,52,63,76,91,108,57,67,

%T 79,93,109,127,147,73,84,97,112,129,148,169,192,91,103,117,133,151,

%U 171,193,217,243,111,124,139,156,175,196,219,244,271,300,133,147,163

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

%H Reinhard Zumkeller, <a href="/A215631/b215631.txt">Rows n = 1..120 of triangle, flattened</a>

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

%F 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

%F From _Avi Friedlich_, May 26 2015: (Start)

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

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

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

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

%e The triangle begins:

%e row n   T(n,k), 1 <= k <= n

%e    1:     3

%e    2:     7   12

%e    3:    13   19   27

%e    4:    21   28   37   48

%e    5:    31   39   49   61   75

%e    6:    43   52   63   76   91  108

%e    7:    57   67   79   93  109  127  147

%e    8:    73   84   97  112  129  148  169  192

%e    9:    91  103  117  133  151  171  193  217  243

%e   10:   111  124  139  156  175  196  219  244  271  300

%e   11:   133  147  163  181  201  223  247  273  301  331  363

%e   12:   157  172  189  208  229  252  277  304  333  364  397  432

%p seq(seq(i^2+i*j+j^2, j=1..i),i=1..10); # _Robert Israel_, May 10 2015

%t Table[n^2 + n*k + k^2, {n, 11}, {k, n}] // Flatten (* _Michael De Vlieger_, May 12 2015 *)

%o (Haskell)

%o a215631 n k = a215631_tabl !! (n-1) !! (k-1)

%o a215631_row n = a215631_tabl !! (n-1)

%o a215631_tabl = zipWith3 (zipWith3 (\u v w -> u + v + w))

%o                         a093995_tabl a075362_tabl a133819_tabl

%o -- _Reinhard Zumkeller_, Nov 11 2012

%o (PARI) for(n=1,15,for(k=1,n,print1(n^2+n*k+k^2,", "))) \\ _Derek Orr_, May 13 2015

%o (Magma) [[i^2+i*j+j^2: j in [1..i]]: i in [1..10]]; // _Vincenzo Librandi_, Jun 07 2015

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

%K nonn,tabl,easy

%O 1,1

%A _Reinhard Zumkeller_, Nov 11 2012