A290132 The number of edges in a graph induced by a regular drawing of K_{n,n}.

1, 6, 24, 74, 170, 362, 642, 1110, 1766, 2706, 3894, 5558, 7602, 10326, 13562, 17510, 22178, 28006, 34634, 42722, 51922, 62570, 74450, 88462, 103994, 121862, 141482, 163610, 187886, 215578, 245430, 279198, 315958, 356390, 399830, 447542, 498626, 555278, 615698, 681206

Offset: 1

Author

R. J. Mathar, Jul 20 2017

Keywords

• nonn
• easy

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5, Table 2.

Crossrefs

Cf. A159065, A290131.

Programs

Maple

A290132 := proc(n)
    2*n+A290131(n)+A159065(n)-1 ;
end proc:
seq(A290132(n),n=1..40);

Mathematica

b[n_] := Sum[(n-i+1)(n-j+1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
A290131[n_] := b[n-1] + (n-1)^2;
A159065[n_] := Module[{x, y, s1 = 0, s2 = 0}, For[x = 1, x <= n - 1, x++, For[y = 1, y <= n - 1, y++, If[GCD[x, y] == 1, s1 += (n - x)(n - y); If[2x <= n - 1 && 2y <= n - 1, s2 += (n - 2x)(n - 2y)]]]]; s1 - s2];
a[n_] := 2n + A290131[n] + A159065[n] - 1;
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, May 24 2023, after Joerg Arndt in A159065 *)

Python

from math import gcd
def a115004(n):
    r=0
    for a in range(1, n + 1):
        for b in range(1, n + 1):
            if gcd(a, b)==1:r+=(n + 1 - a)*(n + 1 - b)
    return r
def a159065(n):
    c=0
    for a in range(1, n):
        for b in range(1, n):
            if gcd(a, b)==1:
                c+=(n - a)*(n - b)
                if 2*aIndranil Ghosh, Jul 20 2017

Formula

a(n) = 2*n + A290131(n) + A159065(n) - 1.