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A255250 Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m. 2

%I

%S 1,2,3,1,4,2,1,5,3,2,6,4,3,2,7,5,4,3,1,8,6,5,4,2,9,7,6,5,3,2,10,8,7,6,

%T 5,3,1,11,9,8,7,6,4,3,1,12,10,9,8,7,5,4,2,13,11,10,9,8,6,5,3,1,14,12,

%U 11,10,9,8,6,4,3,1,15,13,12,11,10,9,7,6,4,2,16,14,13,12,11,10,8,7,5,4,2

%N Array T(n, m) of numbers of points of a square lattice in the first octant covered by a circular disk of radius n (centered at any lattice point taken as origin) with ordinate y = m.

%C The row length of this array (irregular triangle) is 1 + flpoor(n/sqrt(2)) = 1 + A049472(n) = 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 10, 11, ...

%C This entry is motivated by the proposal A255195 by Mats Granvik, who gave the first differences of this array.

%C See the MathWorld link on Gauss's circle problem.

%C The first octant of a square lattice (x, y) with n = x >= y = m >= 0 is considered. The number of lattice points in this octant covered by a circular disk of radius R = n around the origin having ordinate value y = m are denoted by T(n, m), for n >= 0 and m= 0, 1,..., floor(n/sqrt(2)).

%C The row sums give RS(n) = A036702(n), n >= 0. This is the total number of square lattice points in the first octant covered by a circular disk of radius R = n.

%C The alternating row sums give A256094(n), n >= 0.

%C The total number of square lattice points in the first quadrant covered by a circular disk of radius R = n is therefore 2*RS(n) - (1 + floor(n/sqrt(2))) = A000603(n).

%H E. W. Weisstein, World of Mathematics, <a href="http://mathworld.wolfram.com/GausssCircleProblem.html">Gauss's Circle Problem </a>.

%F T(n, m) = floor(sqrt(n^2 - m^2)) - (m-1), n >= 0, m = 0,1, ..., floor(n/sqrt(2)).

%e The array (irregular triangle) T(n, m) begins:

%e n\m 0 1 2 3 4 5 6 7 8 9 10 ....

%e 0: 1

%e 1: 2

%e 2: 3 1

%e 3: 4 2 1

%e 4: 5 3 2

%e 5: 6 4 3 2

%e 6: 7 5 4 3 1

%e 7: 8 6 5 4 2

%e 8: 9 7 6 5 3 2

%e 9: 10 8 7 6 5 3 1

%e 10: 11 9 8 7 6 4 3 1

%e 11: 12 10 9 8 7 5 4 2

%e 12: 13 11 10 9 8 6 5 3 1

%e 13: 14 12 11 10 9 8 6 4 3 1

%e 14: 15 13 12 11 10 9 7 6 4 2

%e 15: 16 14 13 12 11 10 8 7 5 4 2

%e ...

%Y Cf. A036702, A256094, A000328, A049472, A255195, A255238 (quadrant).

%K nonn,easy,tabf

%O 0,2

%A _Wolfdieter Lang_, Mar 14 2015

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Last modified October 13 16:43 EDT 2019. Contains 327968 sequences. (Running on oeis4.)