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A115004 Main diagonal of array in A114999. 29
1, 8, 31, 80, 179, 332, 585, 948, 1463, 2136, 3065, 4216, 5729, 7568, 9797, 12456, 15737, 19520, 24087, 29308, 35315, 42120, 50073, 58920, 69025, 80264, 92871, 106756, 122475, 139528, 158681, 179608, 202529, 227400, 254597, 283784, 315957, 350576, 387977 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Also (1/4) * number of ways to select 3 distinct points forming a triangle of unsigned area = 1/2 from a square of grid points with side length n. Diagonal of triangle A320541. - Hugo Pfoertner, Oct 22 2018

LINKS

Ray Chandler, Table of n, a(n) for n = 1..1000

M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5.

S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph , JIS 12 (2009) 09.5.5.

N. J. A. Sloane (in collaboration with Scott R. Shannon), Art and Sequences, Slides of guest lecture in Math 640, Rutgers Univ., Feb 8, 2020. Mentions this sequence.

FORMULA

a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j).

MAPLE

A115004 := 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:

seq(A115004(n), n=1..30); # R. J. Mathar, Jul 20 2017

MATHEMATICA

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

Array[a, 40] (* Jean-Fran├žois Alcover, Mar 23 2020 *)

PROG

(Python)

from fractions 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

print map(a115004, range(1, 51)) # Indranil Ghosh, Jul 21 2017

CROSSREFS

Cf. A320540, A320541, A320544.

The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. - N. J. A. Sloane, Feb 04 2020

Sequence in context: A240707 A115293 A212579 * A303522 A299261 A005338

Adjacent sequences:  A115001 A115002 A115003 * A115005 A115006 A115007

KEYWORD

nonn,nice,changed

AUTHOR

N. J. A. Sloane, Feb 23 2006

STATUS

approved

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Last modified April 3 19:35 EDT 2020. Contains 333198 sequences. (Running on oeis4.)