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A279413 Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k. 10

%I

%S 0,0,4,0,2,12,0,0,6,16,0,2,4,6,24,0,0,2,8,10,28,0,2,4,2,8,6,36,0,0,2,

%T 0,6,8,10,40,0,2,4,2,12,10,8,10,56,0,0,2,4,2,4,10,8,10,60,0,2,4,2,4,2,

%U 12,6,12,6,60,0,0,2,0,2,4,6,12,6,8,14,64,0,2

%N Triangle read by rows: T(n,k), n>=k>=1, is the number of isosceles triangles with integer coordinates that have a bounding box of size n X k.

%H Lars Blomberg, <a href="/A279413/b279413.txt">Table of n, a(n) for n = 1..9870</a> (the first 140 rows)

%H Lars Blomberg, <a href="/A279413/a279413.pdf">Algorithms for computing A279413, A279414, A186434 and A271908</a>

%e Triangle begins:

%e 0

%e 0, 4

%e 0, 2, 12

%e 0, 0, 6, 16

%e 0, 2, 4, 6, 24

%e 0, 0, 2, 8, 10, 28

%e 0, 2, 4, 2, 8, 6, 36

%e 0, 0, 2, 0, 6, 8, 10, 40

%e 0, 2, 4, 2, 12, 10, 8, 10, 56

%e 0, 0, 2, 4, 2, 4, 10, 8, 10, 60

%e 0, 2, 4, 2, 4, 2, 12, 6, 12, 6, 60

%e 0, 0, 2, 0, 2, 4, 6, 12, 6, 8, 14, 64

%e 0, 2, 4, 2, 4, 6, 8, 10, 16, 14, 12, 14, 72

%e 0, 0, 2, 0, 2, 4, 2, 8, 14, 4, 6, 12, 18, 76

%e 0, 2, 4, 2, 4, 2, 8, 2, 8, 10, 16, 10, 12, 10, 84

%e 0, 0, 2, 0, 6, 4, 2, 4, 6, 16, 6, 4, 10, 12, 14, 88

%e 0, 2, 4, 2, 4, 2, 8, 2, 16, 6, 16, 10, 16, 6, 24, 10, 104

%e 0, 0, 2, 0, 2, 0, 2, 4, 6, 4, 10, 12, 10, 12, 10, 12, 14, 100

%e 0, 2, 4, 2, 4, 2, 12, 6, 4, 6, 12, 10, 20, 6, 12, 14, 16, 10, 124

%e 0, 0, 2, 0, 2, 0, 2, 0, 2, 4, 6, 12, 10, 12, 10, 12, 18, 12, 10, 112

%e -----

%e Denote by 'o' the point adjacent to the two equal sides, and by 'x' the other two.

%e n=4, k=3:

%e ...x x... .o.. ..o. x... ...x

%e o... ...o ...x x... ...x x...

%e ...x x... x... ...x .o.. ..o.

%e So T(4,3)=6.

%e -----

%e n=4,k=4:

%e o... ...o .x.. ..x. o... ...o ..x. .x..

%e ...x x... .... .... .... .... ...x x...

%e .... .... ...x x... ...x x... .... ....

%e .x.. ..x. o... ...o ..x. .x.. o... ...o

%e -

%e ...x x... x... ...x o..x x..o x... ...x

%e .o.. ..o. .... .... .... .... .... ....

%e .... .... .o.. ..o. .... .... .... ....

%e x... ...x ...x x... x... ...x o..x x..o

%e So T(4,4)=16.

%Y Cf. A186434, A187452, A271910-A271913, A271915, A279414.

%Y See A279415 for right isosceles triangles.

%Y See A280639 for obtuse isosceles triangles.

%Y See A279418 for acute isosceles triangles.

%Y See A279433 for all right triangles.

%Y See A280652 for all obtuse triangles.

%Y See A280653 for all acute triangles.

%Y See A279432 for all triangles.

%K nonn,tabl

%O 1,3

%A _Lars Blomberg_, Feb 16 2017

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Last modified January 18 16:27 EST 2020. Contains 331011 sequences. (Running on oeis4.)