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 A130684 Triangle read by rows: T(n,k) = number of squares (not necessarily orthogonal) all of whose vertices lie in an (n + 1) X (k + 1) square lattice. 2
 1, 2, 6, 3, 10, 20, 4, 14, 30, 50, 5, 18, 40, 70, 105, 6, 22, 50, 90, 140, 196, 7, 26, 60, 110, 175, 252, 336, 8, 30, 70, 130, 210, 308, 420, 540, 9, 34, 80, 150, 245, 364, 504, 660, 825, 10, 38, 90, 170, 280, 420, 588, 780, 990, 1210, 11, 42, 100, 190, 315, 476, 672 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Reading down the diagonal gives A002415. LINKS Joel B. Lewis, Jun 29 2007, Table of n, a(n) for n = 1..210 Problem solved on the Art of Problem Solving forum, Number of squares in a grid FORMULA T(n, k) = k*(k+1)*(k+2)*(2*n - k + 1)/12 (k <= n). EXAMPLE T(2, 2) = 6 because there are 6 squares all of whose vertices lie in a 3 X 3 lattice: four squares of side length 1, one square of side length 2 and one non-orthogonal square of side length the square root of 2. Triangle begins: 1; 2, 6; 3, 10, 20; 4, 14, 30, 50; 5, 18, 40, 70, 105; 6, 22, 50, 90, 140, 196; 7, 26, 60, 110, 175, 252, 336; ... PROG (PARI) T(n, k) = binomial(k+2, 3)*(2*n - k + 1)/2 \\ Charles R Greathouse IV, Mar 08 2017 CROSSREFS Cf. A002415. For squares whose edges are required to be parallel to the edges of the large square, see A082652. Sequence in context: A276578 A277810 A180428 * A295644 A079178 A322365 Adjacent sequences: A130681 A130682 A130683 * A130685 A130686 A130687 KEYWORD easy,nonn,tabl AUTHOR Joel B. Lewis, Jun 29 2007 STATUS approved

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Last modified December 4 21:08 EST 2023. Contains 367565 sequences. (Running on oeis4.)