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!)
A331776 Number of regions in a "frame" of size n X n (see Comments for definition). 14
4, 56, 208, 496, 1016, 1784, 2984, 4656, 6968, 9944, 13976, 18928, 25360, 33128, 42488, 53600, 67232, 82904, 101744, 123232, 147896, 175784, 208296, 244416, 285600, 331352, 382608, 439008, 502776, 571912, 649480, 734176, 826880, 927416, 1037288, 1155152, 1284992 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
A "frame" of size n X n is formed from a grid of (n+1) X (n+1) points with the central grid of (n-3) X (n-3) points removed. If n is less than 3 then no points are removed, and a(n) = A255011(n). From now on we assume n >= 3.
If we focus on the squares rather than the points, the frame consists of an n X n array of squares with the central block of (n-2) X (n-2) squares removed.
The resulting structure has an outer perimeter with 4*n points and an inner perimeter with 4*n-8 points, for a total of 8*n-8 perimeter points. The frame itself is the strip of width 1 between the inner and outer perimeters.
Now join every pair of perimeter points, both inner and outer, by a line segment, provided the line remains inside the frame. The sequence gives the number of regions in the resulting figure.
Theorem. Let z(n) = Sum_{i, j = 1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) (this is A115004). Then, for n >= 2, a(n) = 4*z(n) + 16*n^2 - 20*n. - Scott R. Shannon and N. J. A. Sloane, Mar 06 2020
LINKS
Scott R. Shannon, Colored illustration for a(8) = 4656 (Another version)
FORMULA
For n > 1, a(n) = 20*n*(n-1) + 4*Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021
MAPLE
# First define z(n) = A115004
z := proc(n)
local a, b, r ;
r := 0 ;
for a from 1 to n do
for b from 1 to n do
if igcd(a, b) = 1 then
r := r+(n+1-a)*(n+1-b);
end if;
end do:
end do:
r ;
end proc:
A331776 := n -> if n=1 then 4 else 4*z(n)+16*n^2 - 20*n; fi;
[seq(A331776(n), n=1..40)]; # N. J. A. Sloane, Mar 09 2020
PROG
(PARI) a(n) = 4*sum(i=1, n, sum(j=1, n, if(gcd(i, j)==1, (n+1-i)*(n+1-j), 0))) + 16*n^2 - 20*n + 4*(n==1); \\ Jinyuan Wang, Aug 07 2021
(Python)
from sympy import totient
def A331776(n): return 4 if n == 1 else 20*n*(n-1) + 4*sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2, n+1)) # Chai Wah Wu, Aug 16 2021
CROSSREFS
This is the main diagonal of A331457. Equals 4 times A332594.
The analogous sequence for an n X n block of squares (if the center block is not removed) is A331452.
Sequence in context: A360262 A192845 A144138 * A006592 A201448 A195577
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from N. J. A. Sloane, Mar 09 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 March 19 02:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)