cp's OEIS Frontend

A248333 Number of unit squares enclosed by n lattice points in and along the first quadrant of the coordinate plane starting from (0,0) and moving along each square gnomon starting on the y-axis and ending on the x-axis.

%I A248333 #17 Mar 21 2021 12:55:58
%S A248333 0,0,0,0,1,1,2,2,3,4,4,5,6,6,7,8,9,9,10,11,12,12,13,14,15,16,16,17,18,
%T A248333 19,20,20,21,22,23,24,25,25,26,27,28,29,30,30,31,32,33,34,35,36,36,37,
%U A248333 38,39,40,41,42,42,43,44,45,46,47,48,49,49,50,51,52,53,54,55,56,56,57,58,59
%N A248333 Number of unit squares enclosed by n lattice points in and along the first quadrant of the coordinate plane starting from (0,0) and moving along each square gnomon starting on the y-axis and ending on the x-axis.
%C A248333 For n > 0, the pattern fails to add a square if n is of the form k^2+1 (A002522) or k^2-k+1 (A002061). Taken together, these numbers are: 1, 2, 3, 5, 7, 10, 13, 17, ... All other numbers add one unit square to the pattern (see example).
%e A248333 Figure 1:                                                            .
%e A248333                                  .     . .   . . .   . . .   . . .   . . .
%e A248333                  .    . .  . .   . .   . .   . .     . . .   . . .   . . .
%e A248333             .    .    .    . .   . .   . .   . .     . .     . . .   . . .
%e A248333        -------------------------------------------------------------------
%e A248333 n:     0    1    2    3    4     5     6     7       8       9       10
%e A248333 a(n):  0    0    0    0    1     1     2     2       3       4       4
%e A248333 --                                                             .
%e A248333        . .     . . .   . . . .   . . . .   . . . .   . . . .   . . . .
%e A248333        . . .   . . .   . . .     . . . .   . . . .   . . . .   . . . .
%e A248333        . . .   . . .   . . .     . . .     . . . .   . . . .   . . . .
%e A248333        . . .   . . .   . . .     . . .     . . .     . . . .   . . . .
%e A248333        -------------------------------------------------------------------
%e A248333 n:     11      12      13        14        15        16        17
%e A248333 a(n):  5       6       6         7         8         9         9
%e A248333 --
%e A248333 n = 4 -->(0,0), (0,1), (1,1), (1,0) enclose one unit square.
%e A248333 n = 5 -->(0,0), (0,1), (1,1), (1,0), (0,2) also enclose one unit square.
%e A248333 n = 6 -->(0,0), (0,1), (1,1), (1,0), (0,2), (1,2) enclose two unit squares.
%p A248333 A248333:=n->add(`if`(issqr(i-1) or issqr(floor(i-sqrt(i-1))),0,1),i=1..n): seq(A248333(n), n=0..100);
%o A248333 (PARI) a(n) = sum(i=1, n, !(issquare(i-1) || issquare(floor(i-sqrt(i-1))))); \\ _Michel Marcus_, Mar 21 2021
%Y A248333 Cf. A002061 (n^2-n+1), A002522 (n^2+1).
%K A248333 nonn
%O A248333 0,7
%A A248333 _Wesley Ivan Hurt_, Oct 04 2014