A332350
Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.
Original entry on oeis.org
0, 2, 12, 4, 26, 56, 6, 44, 98, 172, 8, 66, 148, 262, 400, 10, 92, 210, 376, 578, 836, 12, 122, 280, 502, 772, 1118, 1496, 14, 156, 362, 652, 1006, 1460, 1958, 2564, 16, 194, 452, 818, 1264, 1838, 2468, 3234, 4080, 18, 236, 554, 1004, 1554, 2264, 3042, 3988, 5034, 6212
Offset: 1
Triangle begins:
0,
2, 12,
4, 26, 56,
6, 44, 98, 172,
8, 66, 148, 262, 400,
10, 92, 210, 376, 578, 836,
12, 122, 280, 502, 772, 1118, 1496,
14, 156, 362, 652, 1006, 1460, 1958, 2564,
16, 194, 452, 818, 1264, 1838, 2468, 3234, 4080,
18, 236, 554, 1004, 1554, 2264, 3042, 3988, 5034, 6212,
...
- Max A. Alekseyev, On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631. doi:10.1137/090750184. See N(m,n) in Theorem 2.
- M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090. This sequence is f_1(m,n).
-
VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
for m from 1 to 12 do lprint(seq(VR(m,n,1),n=1..m),); od:
-
T[m_, n_] := Sum[Boole[GCD[i, j] == 1] (m - Abs[i]) (n - Abs[j]), {i, -m + 1, m - 1}, {j, -n + 1, n - 1}];
Table[T[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Apr 19 2020 *)
A332612
a(n) = Sum_{ i=2..n-1, j=1..i-1, gcd(i,j)=1 } (n-i)*(n-j).
Original entry on oeis.org
0, 0, 2, 11, 32, 77, 148, 268, 442, 691, 1018, 1472, 2036, 2780, 3686, 4786, 6100, 7724, 9598, 11863, 14454, 17437, 20818, 24772, 29172, 34200, 39794, 46071, 52986, 60817, 69314, 78860, 89292, 100720, 113122, 126686, 141244, 157294, 174566, 193228, 213172, 234954, 258058, 283189, 309946, 338473, 368782, 401516, 436040
Offset: 1
The following eight sequences are all essentially the same. The simplest is
A115004(n), which we denote by z(n). Then
A088658(n) = 4*z(n-1);
A114043(n) = 2*z(n-1)+2*n^2-2*n+1;
A114146(n) = 2*
A114043(n);
A115005(n) = z(n-1)+n*(n-1);
A141255(n) = 2*z(n-1)+2*n*(n-1);
A290131(n) = z(n-1)+(n-1)^2;
A306302(n) = z(n)+n^2+2*n. The present sequence and
A331771 could be added to this list.
-
I1 := proc(n) local a, i, j; a:=0;
for i from 2 to n-1 do for j from 1 to i-1 do
if igcd(i,j)=1 then a := a+(n-i)*(n-j); fi; od; od; a; end;
[seq(I1(n),n=1..40)];
-
a(n) = sum(i=2, n-1, sum(j=1, i-1, if (gcd(i,j)==1, (n-i)*(n-j)))); \\ Michel Marcus, Mar 14 2020
-
from sympy import totient
def A332612(n): return sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n))//2 # Chai Wah Wu, Aug 17 2021
A339400
Mark each point on the n X n grid with the number of points that are visible from it; a(n) is the number of distinct values in the grid.
Original entry on oeis.org
1, 3, 3, 4, 3, 7, 5, 7, 7, 11, 5, 14, 8, 13, 13, 19, 9, 22, 11, 23, 21, 25, 13, 29, 21, 34, 26, 37, 11, 40, 26, 44, 31, 45, 21, 54, 35, 54, 36, 55, 24, 65, 40, 59, 47, 70, 24, 71, 43, 72, 55, 81, 28, 74, 55, 88, 59, 90, 28, 93, 58, 91, 66, 96, 46, 110, 63, 100
Offset: 1
a(1) = 1 because there are 3 visible points from every point on the grid.
a(2) = 3 because 5 points are visible from every vertex of the grid, 7 points are visible from the midpoint of every edge of the grid, and 8 points are visible from the middle of the grid.
a(3) = 3 because 9 points are visible from every vertex of the grid, 11 points are visible from the inner points of every edge of the grid, and 12 points are visible from every inner point of the grid.
-
\\ n = side length, d = dimension
cdvps(n, d) ={my(m=Map());
forvec(u=vector(d, i, [0, n\2]),
my(c=0); forvec(v=[[t-n, t]|t<-u], c+=(gcd(v)==1));
mapput(m, c, 1), 1);
#m; }
a(n) = cdvps(n, 2)
Showing 1-3 of 3 results.
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