A373710 Triangle read by rows: T(n,k) is the area of the square whose vertices divide the sides n of a circumscribed square into integer sections k and n - k, 0 <= k <= floor(n/2).

0, 1, 4, 2, 9, 5, 16, 10, 8, 25, 17, 13, 36, 26, 20, 18, 49, 37, 29, 25, 64, 50, 40, 34, 32, 81, 65, 53, 45, 41, 100, 82, 68, 58, 52, 50, 121, 101, 85, 73, 65, 61, 144, 122, 104, 90, 80, 74, 72, 169, 145, 125, 109, 97, 89, 85, 196, 170, 148, 130, 116, 106, 100, 98

Offset: 0

Author

Felix Huber, Jun 17 2024

Keywords

• nonn
• tabf
• easy

Comments

For a sketch see linked illustration "Square in square".

Examples

			Triangle T(n,k) begins:
   n\k   0     1     2     3     4     5     6     7   ...
   0     0
   1     1
   2     4     2
   3     9     5
   4    16    10     8
   5    25    17    13
   6    36    26    20    18
   7    49    37    29    25
   8    64    50    40    34    32
   9    81    65    53    45    41
  10   100    82    68    58    52    50
  11   121   101    85    73    65    61
  12   144   122   104    90    80    74    72
  13   169   145   125   109    97    89    85
  14   196   170   148   130   116   106   100    98
  ...
		

Links

• Felix Huber, Table of n, a(n) for n = 0..100000
• Felix Huber, Square in square

Crossrefs

Cf. A000290(first column), A005563 (second column), A048147 (rows: first half of each diagonal there), A087475 (third column), A189834 (fourth column), A241751 (fifth column).

Programs

Maple

A373710:=(n,k)->n^2+2*k^2-2*n*k;
seq(seq(A373710(n,k),k=0..floor(n/2)),n=0..14);

Formula

T(n,k) = n^2 + 2*k^2 - 2*n*k, 0 <= k <= floor(n/2).

Sequence of row n = r: a(i) = 2*i^2 - 4*i - 2*r*i + r^2 + 2*r + 2, 1 <= i <= floor(r/2 + 1).

Sequence of column k = c: a(j) = j^2 - 2*j + 2*c*j + 2*c^2 - 2*c + 1, j >= 1.