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.

Showing 1-10 of 16 results. Next

A191667 Dispersion of A016813 (4k+1, k>1), by antidiagonals.

Original entry on oeis.org

1, 5, 2, 21, 9, 3, 85, 37, 13, 4, 341, 149, 53, 17, 6, 1365, 597, 213, 69, 25, 7, 5461, 2389, 853, 277, 101, 29, 8, 21845, 9557, 3413, 1109, 405, 117, 33, 10, 87381, 38229, 13653, 4437, 1621, 469, 133, 41, 11, 349525, 152917, 54613, 17749, 6485, 1877, 533
Offset: 1

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

Examples

			Northwest corner:
1....5....21....85....341
2....9....37....149...597
3....13...53....213...853
4....17...69....277...1109
6....25...101...405...1621

Links

Crossrefs

Row 1: A002450.
Cf. A004772, A016813, A191671, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 4*n+1
Table[f[n], {n, 1, 30}]  (* A016813 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191667 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191667  *)

A191702 Dispersion of A008587 (5,10,15,20,25,30,...), by antidiagonals.

Original entry on oeis.org

1, 5, 2, 25, 10, 3, 125, 50, 15, 4, 625, 250, 75, 20, 6, 3125, 1250, 375, 100, 30, 7, 15625, 6250, 1875, 500, 150, 35, 8, 78125, 31250, 9375, 2500, 750, 175, 40, 9, 390625, 156250, 46875, 12500, 3750, 875, 200, 45, 11, 1953125, 781250, 234375, 62500, 18750
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type
"(a or b or c or d mod m)", used in Mathematica programs for A191707-A191711): if f(n)=(n mod 3), then
(a,b,c,d,a,b,c,d,a,b,c,d,...) is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n), so that for n>=1,
"(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
  1...5....25....125...625
  2...10...50....250...1250
  3...15...75....375...1875
  4...20...100...500...2500
  6...30...150...750...3750

Links

Crossrefs

Cf. A047201, A008587, A191707, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n
Table[f[n], {n, 1, 30}]  (* A008587 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]]
(* A191702 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191702  *)

Formula

T(i,j) = T(i,1)*T(1,j) = (i-1+floor((i+3)/4))*5^(j-1), i>=1, j>=1.

A191673 Dispersion of A004773 (>1 and congruent to 0 or 1 or 2 mod 4), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 6, 8, 10, 11, 9, 12, 14, 16, 15, 13, 17, 20, 22, 21, 19, 18, 24, 28, 30, 29, 26, 23, 25, 33, 38, 41, 40, 36, 32, 27, 34, 45, 52, 56, 54, 49, 44, 37, 31, 46, 61, 70, 76, 73, 66, 60, 50, 42, 35, 62, 82, 94, 102, 98, 89, 81, 68, 57, 48, 39, 84
Offset: 1

Views

Author

Clark Kimberling, Jun 11 2011

Keywords

Comments

For a background discussion of dispersions, see A191426.
...
Each of the sequences (4n, n>2), (4n+1, n>0), (3n+2, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The six sequences and dispersions are listed here:
...
A191452=dispersion of A008586 (4k, k>=1)
A191667=dispersion of A016813 (4k+1, k>=1)
A191668=dispersion of A016825 (4k+2, k>=0)
A191669=dispersion of A004767 (4k+3, k>=0)
A191670=dispersion of A042968 (1 or 2 or 3 mod 4 and >=2)
A191671=dispersion of A004772 (0 or 1 or 3 mod 4 and >=2)
A191672=dispersion of A004773 (0 or 1 or 2 mod 4 and >=2)
A191673=dispersion of A004773 (0 or 2 or 3 mod 4 and >=2)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191452 has 1st col A042968, all else A008486
A191667 has 1st col A004772, all else A016813
A191668 has 1st col A042965, all else A016825
A191669 has 1st col A004773, all else A004767
A191670 has 1st col A008486, all else A042968
A191671 has 1st col A016813, all else A004772
A191672 has 1st col A016825, all else A042965
A191673 has 1st col A004767, all else A004773
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c mod m)", (as in the Mathematica program below):
If f(n)=(n mod 3), then (a,b,c,a,b,c,a,b,c,...) is given by
a*f(n+2)+b*f(n+1)+c*f(n), so that "(a or b or c mod m)" is given by
a*f(n+2)+b*f(n+1)+c*f(n)+m*floor((n-1)/3)), for n>=1.

Examples

			Northwest corner:
1....2....4....6....9
3....5....8....12...17
7....10...14...20...28
11...16...22...30...41
15...21...29...40...54

Links

Crossrefs

Cf. A004767, A004773, A191669, A191452.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12; c = 40; c1 = 12;
a = 2; b = 4; c2 = 5; m[n_] := If[Mod[n, 3] == 0, 1, 0];
f[n_] := a*m[n + 2] + b*m[n + 1] + c2*m[n] + 4*Floor[(n - 1)/3]
Table[f[n], {n, 1, 30}] (* A004773 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191673 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191673 *)

A191703 Dispersion of A016861, (5k+1), by antidiagonals.

Original entry on oeis.org

1, 6, 2, 31, 11, 3, 156, 56, 16, 4, 781, 281, 81, 21, 5, 3906, 1406, 406, 106, 26, 7, 19531, 7031, 2031, 531, 131, 36, 8, 97656, 35156, 10156, 2656, 656, 181, 41, 9, 488281, 175781, 50781, 13281, 3281, 906, 206, 46, 10, 2441406, 878906, 253906, 66406, 16406
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1...6... 31....156...781
2...11...56....281...1406
3...16...81....406...2031
4...21...106...531...2656
5...26...131...656...3281
7...36...181...906...4531

Links

Crossrefs

Cf. A047202, A016861, A191708, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n+1
Table[f[n], {n, 1, 30}]  (* A016861 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191703 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191703 *)

A191704 Dispersion of A016873, (5k+2), by antidiagonals.

Original entry on oeis.org

1, 2, 3, 7, 12, 4, 32, 57, 17, 5, 157, 282, 82, 22, 6, 782, 1407, 407, 107, 27, 8, 3907, 7032, 2032, 532, 132, 37, 9, 19532, 35157, 10157, 2657, 657, 182, 42, 10, 97657, 175782, 50782, 13282, 3282, 907, 207, 47, 11, 488282, 878907, 253907, 66407, 16407, 4532
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1...2....7.....32....157
3...12...57....282...1407
4...17...82....407...2032
5...22...107...532...2657
6...27...132...657...3282
6...37...182...907...4532

Links

Crossrefs

Cf. A047207, A016873, A191709, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-3
Table[f[n], {n, 1, 30}] (* A016873 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191704 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191704 *)

A191705 Dispersion of A016873, (5k+3), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 13, 8, 4, 63, 38, 18, 5, 313, 188, 88, 23, 6, 1563, 938, 438, 113, 28, 7, 7813, 4688, 2188, 563, 138, 33, 9, 39063, 23438, 10938, 2813, 688, 163, 43, 10, 195313, 117188, 54688, 14063, 3438, 813, 213, 48, 11, 976563, 585938, 273438, 70313, 17188
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1...3....13....63....313
2...8....38....188...938
4...18...88....438...2188
5...23...113...563...2813
6...28...138...688...3438
7...33...163...813...4063

Links

Crossrefs

Cf. A016885, A032763, A191710, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-2
Table[f[n], {n, 1, 30}] (* A016885 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191705 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191705 *)

A191706 Dispersion of A016873, (5k+4), by antidiagonals.

Original entry on oeis.org

1, 4, 2, 19, 9, 3, 94, 44, 14, 5, 469, 219, 69, 24, 6, 2344, 1094, 344, 119, 29, 7, 11719, 5469, 1719, 594, 144, 34, 8, 58594, 27344, 8594, 2969, 719, 169, 39, 10, 292969, 136719, 42969, 14844, 3594, 844, 194, 49, 11, 1464844, 683594, 214844, 74219, 17969
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1...4....19....94....469
2...9....44....219...1094
3...14...69....344...1719
5...24...119...594...2969
6...29...144...719...3594
7...34...169...844...4219

Links

Crossrefs

Cf. A001068, A016897, A191711, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
f[n_] := 5n-1
Table[f[n], {n, 1, 30}] (* A016897 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191706 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191706 *)

A191707 Dispersion of A016873, (numbers >1 and congruent to 1, 2, 3, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 5, 3, 7, 10, 4, 9, 13, 15, 6, 12, 17, 19, 20, 8, 16, 22, 24, 26, 25, 11, 21, 28, 31, 33, 32, 30, 14, 27, 36, 39, 42, 41, 38, 35, 18, 34, 46, 49, 53, 52, 48, 44, 40, 23, 43, 58, 62, 67, 66, 61, 56, 51, 45, 29, 54, 73, 78, 84, 83, 77, 71, 64, 57, 50, 37
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1....2....3....4....6
5....7....9....12...16
10...13...17...22...28
15...19...24...31...39
20...26...33...42...53
25...32...41...52...66

Links

Crossrefs

Cf. A047201, A008587, A191702, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=3; c2=4; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A047201 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191707 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191707  *)

A191708 Dispersion of A047202, (numbers >1 and congruent to 0, 2, 3, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 2, 6, 3, 8, 11, 4, 10, 14, 16, 5, 13, 18, 20, 21, 7, 17, 23, 25, 27, 26, 9, 22, 29, 32, 34, 33, 31, 12, 28, 37, 40, 43, 42, 39, 36, 15, 35, 47, 50, 54, 53, 49, 45, 41, 19, 44, 59, 63, 68, 67, 62, 57, 52, 46, 24, 55, 74, 79, 85, 84, 78, 72, 65, 58, 51, 30
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1....2....3....4....5
6....8....10...13...17
11...14...18...23...29
16...20...25...32...40
21...27...34...43...54
26...33...42...53...67

Links

Crossrefs

Cf. A047202, A016861, A191703, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=2; b=3; c2=4; d=5; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A047202 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191708 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191708  *)

A191709 Dispersion of A047202, (numbers >1 and congruent to 0, 1, 3, or 4 mod 5), by antidiagonals.

Original entry on oeis.org

1, 3, 2, 5, 4, 7, 8, 6, 10, 12, 11, 9, 14, 16, 17, 15, 13, 19, 21, 23, 22, 20, 18, 25, 28, 30, 29, 27, 26, 24, 33, 36, 39, 38, 35, 32, 34, 31, 43, 46, 50, 49, 45, 41, 37, 44, 40, 55, 59, 64, 63, 58, 53, 48, 42, 56, 51, 70, 75, 81, 80, 74, 68, 61, 54, 47, 71
Offset: 1

Views

Author

Clark Kimberling, Jun 12 2011

Keywords

Comments

For a background discussion of dispersions and their fractal sequences, see A191426. For dispersions of congruence sequences mod 3 or mod 4, see A191655, A191663, A191667.
...
Each of the sequences (5n, n>1), (5n+1, n>1), (5n+2, n>=0), (5n+3, n>=0), (5n+4, n>=0), generates a dispersion. Each complement (beginning with its first term >1) also generates a dispersion. The ten sequences and dispersions are listed here:
...
A191702=dispersion of A008587 (5k, k>=1)
A191703=dispersion of A016861 (5k+1, k>=1)
A191704=dispersion of A016873 (5k+2, k>=0)
A191705=dispersion of A016885 (5k+3, k>=0)
A191706=dispersion of A016897 (5k+4, k>=0)
A191707=dispersion of A047201 (1, 2, 3, 4 mod 5 and >1)
A191708=dispersion of A047202 (0, 2, 3, 4 mod 5 and >1)
A191709=dispersion of A047207 (0, 1, 3, 4 mod 5 and >1)
A191710=dispersion of A032763 (0, 1, 2, 4 mod 5 and >1)
A191711=dispersion of A001068 (0, 1, 2, 3 mod 5 and >1)
...
EXCEPT for at most 2 initial terms (so that column 1 always starts with 1):
A191702 has 1st col A047201, all else A008587
A191703 has 1st col A047202, all else A016861
A191704 has 1st col A047207, all else A016873
A191705 has 1st col A032763, all else A016885
A191706 has 1st col A001068, all else A016897
A191707 has 1st col A008587, all else A047201
A191708 has 1st col A042968, all else A047203
A191709 has 1st col A042968, all else A047207
A191710 has 1st col A042968, all else A032763
A191711 has 1st col A042968, all else A001068
...
Regarding the dispersions A191670-A191673, there is a formula for sequences of the type "(a or b or c or d mod m)", (as in the relevant Mathematica programs):
...
If f(n)=(n mod 3), then (a,b,c,d,a,b,c,d,a,b,c,d,...) is given by a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n); so that for n>=1, "(a, b, c, d mod m)" is given by
a*f(n+3)+b*f(n+2)+c*f(n+1)+d*f(n)+m*floor((n-1)/4)).

Examples

			Northwest corner:
1....3....5....8....11
2....4....6....9....13
7....10...14...19...25
12...16...21...28...36
17...23...30...39...50
22...29...38...49...63

Links

Crossrefs

Cf. A047207, A016873, A191704, A191426.

Programs

  • Mathematica
    (* Program generates the dispersion array T of the increasing sequence f[n] *)
r = 40; r1 = 12;  c = 40; c1 = 12;
a=3; b=4; c2=5; d=6; m[n_]:=If[Mod[n,4]==0,1,0];
f[n_]:=a*m[n+3]+b*m[n+2]+c2*m[n+1]+d*m[n]+5*Floor[(n-1)/4]
Table[f[n], {n, 1, 30}]  (* A047207 *)
mex[list_] := NestWhile[#1 + 1 &, 1, Union[list][[#1]] <= #1 &, 1, Length[Union[list]]]
rows = {NestList[f, 1, c]};
Do[rows = Append[rows, NestList[f, mex[Flatten[rows]], r]], {r}];
t[i_, j_] := rows[[i, j]];
TableForm[Table[t[i, j], {i, 1, 10}, {j, 1, 10}]] (* A191709 *)
Flatten[Table[t[k, n - k + 1], {n, 1, c1}, {k, 1, n}]] (* A191709  *)
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