login
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).
1

%I #12 Jul 07 2024 13:20:04

%S 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 81,65,53,45,41,100,82,68,58,52,50,121,101,85,73,65,61,144,122,104,90,

%U 80,74,72,169,145,125,109,97,89,85,196,170,148,130,116,106,100,98

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

%H Felix Huber, <a href="/A373710/b373710.txt">Table of n, a(n) for n = 0..100000</a>

%H Felix Huber, <a href="/A373710/a373710.pdf">Square in square</a>

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

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

%e Triangle T(n,k) begins:

%e n\k 0 1 2 3 4 5 6 7 ...

%e 0 0

%e 1 1

%e 2 4 2

%e 3 9 5

%e 4 16 10 8

%e 5 25 17 13

%e 6 36 26 20 18

%e 7 49 37 29 25

%e 8 64 50 40 34 32

%e 9 81 65 53 45 41

%e 10 100 82 68 58 52 50

%e 11 121 101 85 73 65 61

%e 12 144 122 104 90 80 74 72

%e 13 169 145 125 109 97 89 85

%e 14 196 170 148 130 116 106 100 98

%e ...

%p A373710:=(n,k)->n^2+2*k^2-2*n*k;

%p seq(seq(A373710(n,k),k=0..floor(n/2)),n=0..14);

%Y 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 nonn,tabf,easy

%O 0,3

%A _Felix Huber_, Jun 17 2024