A132111 Triangle read by rows: T(n,k) = n^2 + k*n + k^2, 0 <= k <= n.

0, 1, 3, 4, 7, 12, 9, 13, 19, 27, 16, 21, 28, 37, 48, 25, 31, 39, 49, 61, 75, 36, 43, 52, 63, 76, 91, 108, 49, 57, 67, 79, 93, 109, 127, 147, 64, 73, 84, 97, 112, 129, 148, 169, 192, 81, 91, 103, 117, 133, 151, 171, 193, 217, 243, 100, 111, 124, 139, 156, 175, 196, 219

Offset

0,3

Comments

Permutation of A003136, the Loeschian numbers. [This is false - some terms are repeated, the first being 49. - Joerg Arndt, Dec 18 2015]

Row sums give A132112.

Central terms give A033582.

T(n,k+1) = T(n,k) + n + 2*k + 1;

T(n+1,k) = T(n,k) + 2*n + k + 1;

T(n+1,k+1) = T(n,k) + 3*(n+k+1);

T(n,0) = A000290(n);

T(n,1) = A002061(n+1) for n>0;

T(n,2) = A117950(n+1) for n>1;

T(n,n-2) = A056107(n-1) for n>1;

T(n,n-1) = A003215(n-1) for n>0;

T(n,n) = A033428(n).

T(n,k) is the norm N(alpha) of the integer alpha = n*1 - k*omega, where omega = exp(2*Pi*i/3) = (-1 + i*sqrt(3))/2 in the imaginary quadratic number field Q(sqrt(-3)): N = |alpha|^2 = (n + k/2)^2 + (3/4)*k^2 = n^2 + n*k + k^2 = T(n,k), with n >= 0, and k <= n. See also triangle A073254 for T(n,-k). - Wolfdieter Lang, Jun 13 2021

Links

Table of n, a(n) for n=0..62.

Example

From Philippe Deléham, Apr 16 2014: (Start)
Triangle begins:
   0;
   1,  3;
   4,  7,  12;
   9, 13,  19,  27;
  16, 21,  28,  37,  48;
  25, 31,  39,  49,  61,  75;
  36, 43,  52,  63,  76,  91, 108;
  49, 57,  67,  79,  93, 109, 127, 147;
  64, 73,  84,  97, 112, 129, 148, 169, 192;
  81, 91, 103, 117, 133, 151, 171, 193, 217, 243;
  ...
(End)

Mathematica

Flatten[Table[n^2+k*n+k^2, {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Jun 10 2013 *)

Crossrefs

Cf. A073254.

Sequence in context: A041153 A041781 A042893 * A164831 A085188 A286728

Adjacent sequences: A132108 A132109 A132110 * A132112 A132113 A132114

Keyword

nonn,tabl,easy

Author

Reinhard Zumkeller, Aug 10 2007

Status

approved