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

0,3

Comments

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

Links

Felix Huber, Table of n, a(n) for n = 0..100000

Felix Huber, Square in square

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.

Example

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
  ...

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);

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).

Sequence in context: A008831 A365378 A289506 * A363268 A213778 A095833

Adjacent sequences: A373707 A373708 A373709 * A373712 A373713 A373714

Keyword

nonn,tabf,easy

Author

Felix Huber, Jun 17 2024

Status

approved