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 A255238 Triangle T(n, m) of numbers of points of a square lattice covered by a circular disk of radius n (centered at any lattice point taken as origin)  with ordinate y = m in the first quadrant. 4
 1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 4, 3, 1, 6, 5, 5, 5, 4, 1, 7, 6, 6, 6, 5, 4, 1, 8, 7, 7, 7, 6, 5, 4, 1, 9, 8, 8, 8, 7, 7, 6, 4, 1, 10, 9, 9, 9, 9, 8, 7, 6, 5, 1, 11, 10, 10, 10, 10, 9, 9, 8, 7, 5, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS This entry is motivated by the proposal A255195 by Mats Granvik. See the MathWorld link on Gauss's circle problem. The first quadrant of a square lattice (x, y) with x = n >= 0, y = m >= 0, is considered. The number of lattice points 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, ..., n. The same numbers occur if x and y are interchanged. One could also consider the row reversed triangle. The row sums give R(n) = A000603(n), n >= 0. The alternating row sums give A255239(n), n >= 0. The total number of square lattice points covered by a circular disk of radius n is A000328(n) = 4*R(n) - (4*n+3). LINKS E. W. Weisstein, World of Mathematics, Gauss's Circle Problem . FORMULA T(n, m) = 1 + floor(sqrt(n^2 - m^2)), 0 <= m <= n. EXAMPLE The triangle T(n, m) begins: n\m  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 0:   1 1:   2  1 2:   3  2  1 3:   4  3  3  1 4:   5  4  4  3  1 5:   6  5  5  5  4  1 6:   7  6  6  6  5  4  1 7:   8  7  7  7  6  5  4  1 8:   9  8  8  8  7  7  6  4  1 9:  10  9  9  9  9  8  7  6  5  1 10: 11 10 10 10 10  9  9  8  7  5  1 11: 12 11 11 11 11 10 10  9  8  7  5  1 12: 13 12 12 12 12 11 11 10  9  8  7  5  1 13: 14 13 13 13 13 13 12 11 11 10  9  7  6  1 14: 15 14 14 14 14 14 13 13 12 11 10  9  8  6  1 15: 16 15 15 15 15 15 14 14 13 13 12 11 10  8  6  1 ... CROSSREFS Cf. A000603, A000328, A255239. Sequence in context: A133334 A003603 A278118 * A212536 A188277 A135227 Adjacent sequences:  A255235 A255236 A255237 * A255239 A255240 A255241 KEYWORD nonn,easy,tabl AUTHOR Wolfdieter Lang, Mar 12 2015 STATUS approved

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Last modified November 27 12:51 EST 2021. Contains 349394 sequences. (Running on oeis4.)