A332598 -id:A332598 - cp's OEIS frontend

A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).

5, 13, 37, 35, 99, 152, 75, 213, 256, 364, 159, 401, 448, 568, 776, 275, 657, 704, 836, 1056, 1340, 477, 1085, 1132, 1276, 1508, 1804, 2272, 755, 1619, 1712, 1868, 2112, 2420, 2900, 3532, 1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336, 1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 03 2020

See A331457 and A331776 for further illustrations.

There is a crucial difference between frames of size nX2 and size nXk with k = 1 or k >= 3. If k != 2, all regions are either triangles or quadrilaterals, but for k=2 regions with larger numbers of sides can appear. Remember also that for k <= 2, the "frame" has no hole, and the graph has genus 0, whereas for k >= 3 there is a nontrivial hole and the graph has genus 1.

			Triangle begins:
[5],
[13, 37],
[35, 99, 152],
[75, 213, 256, 364],
[159, 401, 448, 568, 776],
[275, 657, 704, 836, 1056, 1340],
[477, 1085, 1132, 1276, 1508, 1804, 2272],
[755, 1619, 1712, 1868, 2112, 2420, 2900, 3532],
[1163, 2327, 2552, 2720, 2976, 3296, 3788, 4432, 5336],
[1659, 3257, 3568, 3748, 4016, 4348, 4852, 5508, 6424, 7516],
...

Scott R. Shannon, Colored illustration for T(3,3) = 152
Scott R. Shannon, Colored illustration for T(4,4) = 364
N. J. A. Sloane, Illustration for T(3,3) = 152.

Cf. A331457, A331755, A331763, A331776, A332599, A332600.

The main diagonal is A332598.

Column 1 is A331755, for which there is an explicit formula.
Column 2 is A331763, for which no formula is known.
For m >= n >= 3, T(m,n) = A332600(m,n) - A331457(m,n) (Euler for genus 1 graph), and both A332600 and A331457 have explicit formulas.

More terms from N. J. A. Sloane, Mar 13 2020

A332597 Number of edges in a "frame" of size n X n (see Comments in A331776 for definition).

Original entry on oeis.org

8, 92, 360, 860, 1792, 3124, 5256, 8188, 12304, 17460, 24568, 33244, 44688, 58228, 74664, 94028, 118080, 145380, 178568, 216252, 259776, 308276, 365352, 428556, 501152, 580804, 670536, 768908, 880992, 1001764, 1138248, 1286748, 1449984, 1625300, 1817752, 2023740, 2252048, 2495476, 2759304, 3040460, 3349056

Offset: 1

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

nonn

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, a(n) = 8*z(n) - 4*z_2(n) + 28*n^2 - 44*n + 8. - Scott R. Shannon and N. J. A. Sloane, Mar 06 2020

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Scott R. Shannon, Colored illustration for a(3) = 360 (there are 360 edges in this picture).

Cf. A115004, A331761, A331776 (regions), A332598 (vertices).

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 8 else 28*n^2 - 44*n + 8 + 8*V(n,n,1) - 4*V(n, n, 2); fi;
[seq(f(n), n=1..50)]; # N. J. A. Sloane, Mar 10 2020

from sympy import totient
def A332597(n): return 8 if n == 1 else 4*(n-1)*(8*n-1) + 8*sum(totient(i)*(n+1-i)*(n+i+1) for i in range(2,n//2+1)) + 8*sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1,n+1)) # Chai Wah Wu, Aug 16 2021

For n > 1, a(n) = 4*(n-1)*(8*n-1) + 8*Sum_{i=2..floor(n/2)} (n+1-i)*(n+i+1)*phi(i) + 8*Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

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

cp's OEIS Frontend

A332599 Triangle read by rows: T(n,k) = number of vertices in a "frame" of size n X k (see Comments in A331457 for definition).

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