cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A331776 Number of regions in a "frame" of size n X n (see Comments for definition).

Original entry on oeis.org

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
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Feb 08 2020

Keywords

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

Crossrefs

This is the main diagonal of A331457. Equals 4 times A332594.
Cf. A255011, A115004.
The analogous sequence for an n X n block of squares (if the center block is not removed) is A331452.

Programs

  • 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
  • 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

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

Extensions

More terms from N. J. A. Sloane, Mar 09 2020

A332596 Number of quadrilateral regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 8.

Original entry on oeis.org

0, 1, 10, 26, 63, 107, 189, 294, 455, 627, 891, 1202, 1650, 2121, 2719, 3392, 4292, 5239, 6470, 7832, 9463, 11129, 13205, 15460, 18164, 20919, 24130, 27572, 31679, 35945, 40977, 46340, 52384, 58511, 65421, 72718, 81104, 89589, 98989, 108860, 120062, 131551
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 02 2020

Keywords

Comments

See A331776 for many other illustrations.
Theorem. Let z(n) = Sum_{i, j = 1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) (this is A115004) and z_2(n) = Sum_{i, j = 1..n, gcd(i,j)=2} (n+1-i)*(n+1-j) (this is A331761). Then, for n >= 2, 8*a(n) = 4*z(n) - 8*z_2(n) + 8*n^2 - 36*n + 24. - Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

Links

Crossrefs

Cf. A115004, A331761, A331776, A332594, A332595.

Programs

  • Maple
    V := proc(m, n, q) local a, i, j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i, j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
f := n -> if n=1 then 0 else 8*n^2 - 36*n + 24 + 4*V(n,n,1) 8*V(n, n, 2); fi;
[seq(f(n)/8, n=1..60)]; # N. J. A. Sloane, Mar 10 2020
  • PARI
    a(n) = sum(i=1, n, sum(j=1, n, if(gcd(i, j)==1, (n+1-i)*(n+1-j), 0)))/2 - sum(i=1, n, sum(j=1, n, if(gcd(i, j)==2, (n+1-i)*(n+1-j), 0))) + n^2 - 9*n/2 + 3; \\ Jinyuan Wang, Aug 07 2021
  • Python
    from sympy import totient
def A332596(n): return 0 if n == 1 else ((n-1)*(n-4) - sum(totient(i)*(n+1-i)*(2*n+2-7*i) for i in range(2,n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1,n+1)))//2 # Chai Wah Wu, Aug 16 2021

Formula

For n > 1, a(n) = ((n-1)*(n-4) - Sum_{i=2..floor(n/2)} (n+1-i)*(2*n+2-7*i)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i))/2. - Chai Wah Wu, Aug 16 2021

Extensions

More terms from N. J. A. Sloane, Mar 10 2020

A332595 Number of triangular regions in a "frame" of size n X n (see Comments in A331776 for definition), divided by 4.

Original entry on oeis.org

1, 12, 32, 72, 128, 232, 368, 576, 832, 1232, 1712, 2328, 3040, 4040, 5184, 6616, 8224, 10248, 12496, 15144, 18048, 21688, 25664, 30184, 35072, 41000, 47392, 54608, 62336, 71088, 80416, 90864, 101952, 114832, 128480, 143352, 159040, 176984, 195888, 216424, 237984, 261624, 286384, 313184, 341184, 372496, 405184
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Mar 02 2020

Keywords

Comments

See A331776 for many other illustrations.
Theorem. Let z_2(n) = Sum_{i, j = 1..n, gcd(i,j)=2} (n+1-i)*(n+1-j) (this is A331761). Then, for n >= 2, a(n) = 2*(z_2(n) + (n+3)*(n-1)). - Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

Links

Crossrefs

Cf. A331761, A331776, A332594, A332596.

Programs

  • Maple
    V := proc(m,n,q) local a,i,j; a:=0;
for i from 1 to m do for j from 1 to n do
if gcd(i,j)=q then a:=a+(m+1-i)*(n+1-j); fi; od: od: a; end;
f := n -> if n=1 then 4 else 8*n^2 + 16*n - 24 + 8*V(n,n,2); fi;
[seq(f(n)/4, n=1..60)]; # N. J. A. Sloane, Mar 09 2020

Extensions

More terms from N. J. A. Sloane, Mar 09 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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