A290132 - OEIS

%I #26 May 24 2023 07:33:41

%S 1,6,24,74,170,362,642,1110,1766,2706,3894,5558,7602,10326,13562,

%T 17510,22178,28006,34634,42722,51922,62570,74450,88462,103994,121862,

%U 141482,163610,187886,215578,245430,279198,315958,356390,399830,447542,498626,555278,615698,681206

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

%H Chai Wah Wu, <a href="/A290132/b290132.txt">Table of n, a(n) for n = 1..10000</a>

%H M. Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Griffiths2/griffiths.html">Counting the regions in a regular drawing of K_{n,n}</a>, J. Int. Seq. 13 (2010) # 10.8.5, Table 2.

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

%p A290132 := proc(n)

%p     2*n+A290131(n)+A159065(n)-1 ;

%p end proc:

%p seq(A290132(n),n=1..40);

%t b[n_] := Sum[(n-i+1)(n-j+1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];

%t A290131[n_] := b[n-1] + (n-1)^2;

%t 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];

%t a[n_] := 2n + A290131[n] + A159065[n] - 1;

%t Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, May 24 2023, after _Joerg Arndt_ in A159065 *)

%o (Python)

%o from math import gcd

%o def a115004(n):

%o     r=0

%o     for a in range(1, n + 1):

%o         for b in range(1, n + 1):

%o             if gcd(a, b)==1:r+=(n + 1 - a)*(n + 1 - b)

%o     return r

%o def a159065(n):

%o     c=0

%o     for a in range(1, n):

%o         for b in range(1, n):

%o             if gcd(a, b)==1:

%o                 c+=(n - a)*(n - b)

%o                 if 2*a<n and 2*b<n:c-=(n - 2*a)*(n - 2*b)

%o     return c

%o def a290131(n): return a115004(n - 1) + (n - 1)**2

%o def a(n): return 2*n + a290131(n) + a159065(n) - 1

%o print([a(n) for n in range(1, 51)]) # _Indranil Ghosh_, Jul 20 2017

%Y Cf. A159065, A290131.

%K nonn,easy

%O 1,2

%A _R. J. Mathar_, Jul 20 2017