login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 26 20:03 EDT 2022. Contains 354885 sequences. (Running on oeis4.)