cp's OEIS Frontend

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.

Previous Showing 11-18 of 18 results.

A358304 Array read by antidiagonals: T(n,k) (n>=0, k>=0) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 5, 5, 0, 0, 9, 10, 9, 0, 0, 14, 19, 19, 14, 0, 0, 20, 27, 32, 27, 20, 0, 0, 27, 40, 47, 47, 40, 27, 0, 0, 35, 51, 68, 66, 68, 51, 35, 0, 0, 44, 68, 85, 96, 96, 85, 68, 44, 0, 0, 54, 82, 112, 118, 134, 118, 112, 82, 54, 0, 0, 65, 103, 137, 156, 167, 167, 156, 137, 103, 65, 0, 0, 77, 120, 166, 187, 217, 204, 217, 187, 166, 120, 77, 0
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Examples

			The full array T(n,k), n >= 0, k >= 0, begins:
  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,   0,   0,   0, ..
  0,  2,  5,  9,  14,  20,  27,  35,  44,  54,  65,  77,  90, ..
  0,  5, 10, 19,  27,  40,  51,  68,  82, 103, 120, 145, 165, ..
  0,  9, 19, 32,  47,  68,  85, 112, 137, 166, 196, 235, 265, ..
  0, 14, 27, 47,  66,  96, 118, 156, 187, 229, 266, 320, 358, ..
  0, 20, 40, 68,  96, 134, 167, 217, 261, 317, 366, 436, 491, ..
  0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, 528, 589, ..
  ...

Links

Crossrefs

Cf. A358298.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Programs

  • Maple
    A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m,n) local a,i,j;  # A331781 or equally A333295. Diagonal is A018805.
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
DFD:=proc(m,n) local d,t1,u,v; global A005728, Amn;
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
t1; end;
for m from 0 to 8 do lprint([seq(DFD(m,n),n=0..20)]); od:
  • Mathematica
    T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];
Table[T[n-k, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2023 *)

A358299 Triangle read by antidiagonals: T(n,k) (n>=0, 0 <= k <= n) = number of lines defining the Farey diagram of order (n,k).

Original entry on oeis.org

2, 3, 6, 4, 11, 20, 6, 19, 36, 60, 8, 29, 52, 88, 124, 12, 43, 78, 128, 180, 252, 14, 57, 100, 162, 224, 316, 388, 20, 77, 136, 216, 298, 412, 508, 652, 24, 97, 166, 266, 360, 498, 608, 780, 924, 30, 121, 210, 326, 444, 608, 738, 940, 1116, 1332, 34, 145, 246, 386, 518, 706, 852, 1086, 1280, 1532, 1748
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Examples

			The full array T(n,k), n >= 0, k>= 0, begins:
2, 3, 4, 6, 8, 12, 14, 20, 24, 30, 34, 44, 48, 60, ...
3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205, ...
4, 11, 20, 36, 52, 78, 100, 136, 166, 210, 246, 302, ...
6, 19, 36, 60, 88, 128, 162, 216, 266, 326, 386, 468, ...
8, 29, 52, 88, 124, 180, 224, 298, 360, 444, 518, 628, ...
12, 43, 78, 128, 180, 252, 316, 412, 498, 608, 706, ...
14, 57, 100, 162, 224, 316, 388, 508, 608, 738, 852, ...
...

Links

Crossrefs

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Programs

  • Maple
    A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m,n) local a,i,j; # A331781 or equally A333295. Diagonal is A018805.
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
# The present sequence is:
Dmn:=proc(m,n) local d,t1,u,v,a; global A005728, Amn;
a:=A005728(m)+A005728(n);
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
a+2*t1-2*Amn(m,n); end;
for m from 1 to 8 do lprint([seq(Dmn(m,n),n=1..20)]); od:

A358300 Row 1 of array in A358298.

Original entry on oeis.org

3, 6, 11, 19, 29, 43, 57, 77, 97, 121, 145, 177, 205, 243, 277, 315, 355, 405, 447, 503, 551, 605, 659, 727, 783, 853, 917, 989, 1057, 1143, 1211, 1303, 1383, 1469, 1553, 1647, 1731, 1841, 1935, 2037, 2133, 2255, 2351, 2479, 2587, 2701, 2815, 2955, 3067, 3207, 3327, 3461
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Crossrefs

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

A358301 Main diagonal of array in A358298.

Original entry on oeis.org

2, 6, 20, 60, 124, 252, 388, 652, 924, 1332, 1748, 2428, 2988, 3948, 4788, 5908, 7028, 8692, 9964, 12052, 13748, 16004, 18124, 21204, 23476, 26996, 29972, 33788, 37196, 42124, 45548, 51188, 55732, 61412, 66532, 73348, 78484, 86548, 92956, 100924, 107772, 117692, 124556, 135476, 144036
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Crossrefs

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Programs

  • Mathematica
    A005728[n_] := 1 + Sum[EulerPhi[i], {i, 1, n}];
Amn[m_, n_] := Sum[If[GCD[i, j] == 1, 1, 0], {i, 1, m}, {j, 1, n}];
Dmn[m_, n_] := A005728[m] + A005728[n] + 2 Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, m}, {v, 1, n}] - 2*Amn[m, n];
Table[Dmn[n, n], {n, 0, 44}] (* Jean-François Alcover, Apr 18 2023, after Maple code in A358298 *)

A358302 Number of triangular regions in the Farey Diagram Farey(n,n), divided by 4.

Original entry on oeis.org

1, 12, 100, 392, 1554, 3486, 9690, 18942, 38610, 65268, 125116, 186870, 324646, 472546, 713354, 1003888, 1531908, 2000638, 2920970, 3780950
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Comments

This is the leading column in A358885, divided by 4.
It would be nice to have a formula.

Crossrefs

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

A358303 Number of 4-sided regions in the Farey Diagram Farey(n,n), divided by 8.

Original entry on oeis.org

1, 13, 57, 231, 532, 1497, 2935, 6031, 10273, 19680, 29441, 51261, 74473, 112721, 159299, 242763, 317155, 462930, 598755
Offset: 1

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Comments

This is the second column in A358885, divided by 8.
It would be nice to have a formula.

Crossrefs

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

A358305 Triangle read by rows: T(n,k) (n>=0, 0 <= k <= n) = number of decreasing lines defining the Farey diagram Farey(n,k) of order (n,k).

Original entry on oeis.org

0, 0, 2, 0, 5, 10, 0, 9, 19, 32, 0, 14, 27, 47, 66, 0, 20, 40, 68, 96, 134, 0, 27, 51, 85, 118, 167, 204, 0, 35, 68, 112, 156, 217, 267, 342, 0, 44, 82, 137, 187, 261, 318, 408, 482, 0, 54, 103, 166, 229, 317, 384, 490, 581, 692, 0, 65, 120, 196, 266, 366, 441, 564, 664, 794, 904
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Examples

			The full array T(n,k), n >= 0, k>= 0, begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 2, 5, 9, 14, 20, 27, 35, 44, 54, 65, 77, 90, ...
0, 5, 10, 19, 27, 40, 51, 68, 82, 103, 120, 145, ...
0, 9, 19, 32, 47, 68, 85, 112, 137, 166, 196, 235, ...
0, 14, 27, 47, 66, 96, 118, 156, 187, 229, 266, ...
0, 20, 40, 68, 96, 134, 167, 217, 261, 317, 366, ...
0, 27, 51, 85, 118, 167, 204, 267, 318, 384, 441, ...

Links

Crossrefs

Cf. A358298.
The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.

Programs

  • Maple
    A005728 := proc(n) 1+add(numtheory[phi](i), i=1..n) ; end proc: # called F_n in the paper
Amn:=proc(m,n) local a,i,j; # A331781 or equally A333295. Diagonal is A018805.
a:=0; for i from 1 to m do for j from 1 to n do
if igcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
DFD:=proc(m,n) local d,t1,u,v; global A005728, Amn;
t1:=0; for u from 1 to m do for v from 1 to n do
d:=igcd(u,v); if d>=1 then t1:=t1 + (u+v)*numtheory[phi](d)/d; fi; od: od:
t1; end;
for m from 0 to 8 do lprint([seq(DFD(m,n),n=0..20)]); od:
  • Mathematica
    T[n_, k_] := Sum[d = GCD[u, v]; If[d >= 1, (u+v)*EulerPhi[d]/d, 0], {u, 1, n}, {v, 1, k}];
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 18 2023 *)

A358306 Second row of array in A358304.

Original entry on oeis.org

0, 5, 10, 19, 27, 40, 51, 68, 82, 103, 120, 145, 165, 194, 217, 250, 276, 313, 342, 383, 415, 460, 495, 544, 582, 635, 676, 733, 777, 838, 885, 950, 1000, 1069, 1122, 1195, 1251, 1328, 1387, 1468, 1530, 1615, 1680, 1769, 1837, 1930, 2001, 2098, 2172, 2273, 2350, 2455, 2535, 2644, 2727, 2840, 2926, 3043, 3132, 3253, 3345
Offset: 0

Views

Author

Scott R. Shannon and N. J. A. Sloane, Dec 06 2022

Keywords

Crossrefs

The Farey Diagrams Farey(m,n) are studied in A358298-A358307 and A358882-A358885, the Completed Farey Diagrams of order (m,n) in A358886-A358889.
Previous Showing 11-18 of 18 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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