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.
A290131 := proc(n) A115004(n-1)+(n-1)^2 ; end proc: seq(A290131(n),n=1..30) ;
z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[n_] := z[n - 1] + (n - 1)^2;
Array[a, 40] (* Jean-François Alcover, Mar 24 2020 *)
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 a(n): return a115004(n - 1) + (n - 1)**2
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 20 2017, after Maple code
from sympy import totient def A290131(n): return 2*(n-1)**2 + sum(totient(i)*(n-i)*(2*n-i) for i in range(2,n)) # Chai Wah Wu, Aug 16 2021
# Maple code from N. J. A. Sloane, Jul 16 2020 V106i := proc(n) local ans,a,b; ans:=0; for a from 1 to n-1 do for b from 1 to n-1 do if igcd(a,b)=1 then ans:=ans + (n-a)*(n-b); fi; od: od: ans; end; # A115004 V106ii := proc(n) local ans,a,b; ans:=0; for a from 1 to floor(n/2) do for b from 1 to floor(n/2) do if igcd(a,b)=1 then ans:=ans + (n-2*a)*(n-2*b); fi; od: od: ans; end; # A331761 A331755 := n -> 2*(n+1) + V106i(n+1) - V106ii(n+1);
a[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[2*x<=n-1&&2*y<=n-1,s2+=(n-2*x)*(n-2*y)]]]]; s1-s2]; Table[a[n]+ 2 n, {n, 1, 40}] (* Vincenzo Librandi, Feb 04 2020 *)
Led d denote the number of polygons meeting at a vertex. For n=2, in the interiors of each of the two squares there are 3 points with d=4, and the center point has d=6. So in total there are 6 points with d=4 and 1 with d=6. So row 2 of the triangle is [0, 0, 6, 0, 1]. The triangle begins: 0,0,1, 0,0,6,0,1, 0,0,24,0,2,0,1, 0,0,54,0,8,0,2,0,1, 0,0,124,0,18,0,2,0,2,0,1, 0,0,214,0,32,0,10,0,2,0,2,0,1, 0,0,382,0,50,0,22,0,2,0,2,0,2,0,1, 0,0,598,0,102,0,18,0,12,0,2,0,2,0,2,0,1 ... If we leave out the uninteresting zeros, the triangle begins: [1] [6, 1] [24, 2, 1] [54, 8, 2, 1] [124, 18, 2, 2, 1] [214, 32, 10, 2, 2, 1] [382, 50, 22, 2, 2, 2, 1] [598, 102, 18, 12, 2, 2, 2, 1] [950, 126, 32, 26, 2, 2, 2, 2, 1] [1334, 198, 62, 20, 14, 2, 2, 2, 2, 1] [1912, 286, 100, 10, 30, 2, 2, 2, 2, 2, 1] [2622, 390, 118, 38, 22, 16, 2, 2, 2, 2, 2, 1] ... - _N. J. A. Sloane_, Jul 27 2020
A290132 := proc(n) 2*n+A290131(n)+A159065(n)-1 ; end proc: seq(A290132(n),n=1..40);
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 *)
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
a(1) = 5 as connecting the four vertices of a single rectangle forms one new vertex inside the rectangle, giving a total of 4 + 1 = 5 total intersection points. a(2) = 17 as connecting the six vertices of two adjacent rectangles forms seven vertices inside the rectangles while also forming four vertices outside the rectangles. The total number of intersection points is then 6 + 7 + 4 = 17. See the linked images for further examples.
Triangle begins: 1; 3, 3; 5, 11, 5; 7, 19, 19, 7; 9, 29, 43, 29, 9; 11, 37, 61, 61, 37, 11; 13, 47, 83, 105, 83, 47, 13; 15, 57, 103, 143, 143, 103, 57, 15; 17, 69, 125, 183, 211, 183, 125, 69, 17; 19, 81, 143, 215, 267, 267, 215, 143, 81, 19; 21, 95, 167, 253, 329, 369, 329, 253, 167, 95, 21; 23, 109, 189, 289, 385, 455, 455, 385, 289, 189, 109, 23; 25, 125, 215, 331, 451, 551, 597, 551, 451, 331, 215, 125, 25;
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