The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
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!)
A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2. 6
4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, 40, 180, 320, 520, 720, 1000, 1280, 1620, 1960, 2396 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
2,1
LINKS
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. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3
EXAMPLE
Triangle begins:
4,
8, 20,
12, 32, 52,
16, 48, 80, 124,
20, 64, 108, 168, 228,
24, 84, 144, 228, 312, 428,
28, 104, 180, 288, 396, 544, 692,
32, 128, 224, 360, 496, 684, 872, 1100,
36, 152, 268, 432, 596, 824, 1052, 1328, 1604,
...
MAPLE
# Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015).
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;
VS := proc(m, n) local a, i, j; a:=0; # A331781
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i, j)=1 then a:=a+1; fi; od: od: a; end;
c3 := (m, n) -> VR(m, n, 2)+4; # A332367
for m from 2 to 12 do lprint([seq(c3(m, n), n=2..m)]); od:
[seq(c3(n, n)/4, n=2..40)]; # A332368
c4 := (m, n) -> VR(m, n, 1)/2 - VR(m, n, 2) - 3; # A332369
for m from 2 to 12 do lprint([seq(c4(m, n), n=2..m)]); od:
[seq(c4(n, n), n=2..40)]; # A332370
ct := (m, n) -> c3(m, n)+c4(m, n); # A332371
for m from 2 to 12 do lprint([seq(ct(m, n), n=2..m)]); od:
[seq(ct(n, n), n=2..40)]; # A114043
et := (m, n) -> VR(m, n, 1) - VR(m, n, 2)/2 - VS(m, n) - 2; # A332372
for m from 2 to 12 do lprint([seq(et(m, n), n=2..m)]); od:
[seq(et(n, n), n=2..40)]; # A332373
vt := (m, n) -> et(m, n) - ct(m, n) +1; # A332374
for m from 2 to 12 do lprint([seq(vt(m, n), n=2..m)]); od:
[seq(vt(n, n), n=2..40)]; # A332375
CROSSREFS
For main diagonal see A332368.
Sequence in context: A340948 A265108 A328184 * A273143 A273174 A178447
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Feb 12 2020
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 May 18 19:36 EDT 2024. Contains 372666 sequences. (Running on oeis4.)