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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).

%I A373710 #12 Jul 07 2024 13:20:04
%S A373710 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,
%T A373710 81,65,53,45,41,100,82,68,58,52,50,121,101,85,73,65,61,144,122,104,90,
%U A373710 80,74,72,169,145,125,109,97,89,85,196,170,148,130,116,106,100,98
%N 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).
%C A373710 For a sketch see linked illustration "Square in square".
%H A373710 Felix Huber, <a href="/A373710/b373710.txt">Table of n, a(n) for n = 0..100000</a>
%H A373710 Felix Huber, <a href="/A373710/a373710.pdf">Square in square</a>
%F A373710 T(n,k) = n^2 + 2*k^2 - 2*n*k, 0 <= k <= floor(n/2).
%F A373710 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).
%F A373710 Sequence of column k = c: a(j) = j^2 - 2*j + 2*c*j + 2*c^2 - 2*c + 1, j >= 1.
%e A373710 Triangle T(n,k) begins:
%e A373710    n\k   0     1     2     3     4     5     6     7   ...
%e A373710    0     0
%e A373710    1     1
%e A373710    2     4     2
%e A373710    3     9     5
%e A373710    4    16    10     8
%e A373710    5    25    17    13
%e A373710    6    36    26    20    18
%e A373710    7    49    37    29    25
%e A373710    8    64    50    40    34    32
%e A373710    9    81    65    53    45    41
%e A373710   10   100    82    68    58    52    50
%e A373710   11   121   101    85    73    65    61
%e A373710   12   144   122   104    90    80    74    72
%e A373710   13   169   145   125   109    97    89    85
%e A373710   14   196   170   148   130   116   106   100    98
%e A373710   ...
%p A373710 A373710:=(n,k)->n^2+2*k^2-2*n*k;
%p A373710 seq(seq(A373710(n,k),k=0..floor(n/2)),n=0..14);
%Y A373710 Cf. A000290(first column), A005563 (second column), A048147 (rows: first half of each diagonal there), A087475 (third column), A189834 (fourth column), A241751 (fifth column).
%K A373710 nonn,tabf,easy
%O A373710 0,3
%A A373710 _Felix Huber_, Jun 17 2024