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 44 results. Next

A205842 Numbers k for which 3 divides s(k)-s(j) for some j

Original entry on oeis.org

4, 5, 5, 6, 7, 8, 8, 9, 9, 9, 10, 10, 10, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 20, 20, 20, 21
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(4)-s(2) = 5-2 = 3
s(5)-s(2) = 8-2 = 6
s(5)-s(4) = 8-5 = 3
s(6)-s(1) = 13-1 = 12
s(7)-s(3) = 21-3 = 18
s(8)-s(1) = 34-1 = 33

Crossrefs

Cf. A204892, A205845.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 3; t = d[c]       (* A205841 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]      (* A205842 *)
Table[j[n], {n, 1, z2}]      (* A205843 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205844 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205845 *)

A205845 [s(k)-s(j)]/3, where the pairs (k,j) are given by A205842 and A205843, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 2, 1, 4, 6, 11, 7, 18, 14, 7, 29, 28, 27, 47, 41, 77, 76, 75, 48, 125, 124, 123, 96, 48, 203, 199, 192, 185, 328, 322, 281, 532, 528, 521, 514, 329, 861, 857, 850, 843, 658, 329, 1393, 1392, 1391, 1364, 1316, 1268, 2254, 2248, 2207, 1926, 3648
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.
The first six terms match these differences:
s(4)-s(2) = 5-2 = 3 = 3*1
s(5)-s(2) = 8-2 = 6 = 3*2
s(5)-s(4) = 8-5 = 3 = 3*1
s(6)-s(1) = 13-1 = 12 = 3*4
s(7)-s(3) = 21-3 = 18 = 3*6
s(8)-s(1) = 34-1 = 33 + 3*11
(See the program at A205842.)

Examples

			The first six terms match these differences:
s(4)-s(2) = 5-2 = 3 = 3*1
s(5)-s(2) = 8-2 = 6 = 3*2
s(5)-s(4) = 8-5 = 3 = 3*1
s(6)-s(1) = 13-1 = 12 = 3*4
s(7)-s(3) = 21-3 = 18 = 3*6
s(8)-s(1) = 34-1 = 33 + 3*11

Crossrefs

Cf. A204890, A205842, A205845.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 3; t = d[c]       (* A205841 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]      (* A205842 *)
Table[j[n], {n, 1, z2}]      (* A205843 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205844 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205845 *)

A205847 Numbers k for which 4 divides s(k)-s(j) for some j

Original entry on oeis.org

4, 6, 6, 7, 7, 7, 8, 9, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 19, 19, 19, 19, 19, 20, 20, 20, 21, 21, 21
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(4)-s(1) = 5-1 = 4
s(6)-s(1) = 13-1 = 12
s(6)-s(4) = 13-5 = 8
s(7)-s(1) = 21-1 = 20
s(7)-s(4) = 21-5 = 16
s(7)-s(6) = 21-13 = 8

Crossrefs

Cf. A204892, A205850.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 4; t = d[c]    (* A205846 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]    (* A205847 *)
Table[j[n], {n, 1, z2}]    (* A205848 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)

A205852 Numbers k for which 5 divides s(k)-s(j) for some j

Original entry on oeis.org

5, 6, 6, 7, 9, 10, 11, 11, 12, 12, 12, 13, 14, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 23, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 26, 26, 26, 26, 27, 27, 27
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(5)-s(3) = 8-3 = 5
s(6)-s(3) = 13-3 = 10
s(6)-s(5) = 13-8 = 5
s(7)-s(1) = 21-1 = 20
s(9)-s(4) = 55-5 = 50
s(10)-s(8) = 89-34 = 55

Crossrefs

Cf. A204892, A205855.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]    (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 5; t = d[c]    (* A205851 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}] (* A205852 *)
Table[j[n], {n, 1, z2}] (* A205853 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}](* A205854 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205855 *)

A205857 Numbers k for which 6 divides s(k)-s(j) for some j

Original entry on oeis.org

5, 6, 7, 9, 9, 10, 12, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 27, 27, 27, 27, 28, 28, 28, 28
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(5)-s(2) = 8-2 = 6 = 6*1
s(6)-s(1) = 13-1 = 12 = 6*2
s(7)-s(3) = 21-3 = 18 = 6*3
s(9)-s(1) = 55-1 = 54 = 6*9
s(9)-s(6) = 55-13 = 42 = 6*7
s(10)-s(4) = 89-5 = 84 =6*14

Crossrefs

Cf. A204892, A205857, A205860.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 6; t = d[c]    (* A205856 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]     (* A205857 *)
Table[j[n], {n, 1, z2}]     (* A205858 *)
Table[s[k[n]]-s[j[n]], {n, 1, z2}]    (* A205859 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}]  (* A205860 *)

A205862 Numbers k for which 7 divides s(k)-s(j) for some j

Original entry on oeis.org

5, 8, 9, 9, 10, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 17, 18, 18, 19, 20, 20, 21, 21, 21, 21, 21, 22, 22, 22, 22, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25, 26, 26, 26, 27, 28, 28, 28, 29, 29, 29, 29, 29, 29, 29
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(5)-s(1) = 8-1 = 7 = 7*1
s(8)-s(6) = 34-13 = 21 = 7*3
s(9)-s(6) = 55-13 = 42 = 7*6
s(9)-s(8) = 55-34 = 21 = 7*3
s(10)-s(4) = 89-5 = 84 = 7*12
s(13)-s(6) = 377-13 = 364 =7*52

Crossrefs

Cf. A204892, A205863, A205865.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 500; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]    (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 7; t = d[c]   (* A205861 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]    (* A205862 *)
Table[j[n], {n, 1, z2}]    (* A205863 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205864 *)
Table[(s[k[n]]-s[j[n]])/c, {n,1,z2}] (* A205865 *)

A205867 Numbers k for which 8 divides s(k)-s(j) for some j

Original entry on oeis.org

6, 7, 7, 8, 10, 11, 12, 12, 13, 13, 13, 14, 14, 15, 16, 16, 16, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 19, 20, 20, 20, 21, 22, 22, 22, 22, 23, 23, 23, 24, 24, 24, 24, 24, 25, 25, 25, 25, 25, 25
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(6)-s(4) = 13-5 = 8 = 8*1
s(7)-s(4) = 21-5 = 16 = 8*2
s(7)-s(6) = 21-13 = 8 = 8*1
s(8)-s(2) = 34-2 = 32 = 8*4
s(10)-s(1) = 89-1 = 88 = 8*11
s(11)-s(5) = 144-8 = 136 =8*17

Crossrefs

Cf. A204892, A205868, A205870.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 8; t = d[c]    (* A205866 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]      (* A205867 *)
Table[j[n], {n, 1, z2}]        (* A205868 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205869 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205870 *)

A205872 Numbers k for which 9 divides s(k)-s(j) for some j

Original entry on oeis.org

7, 9, 10, 12, 12, 13, 13, 13, 14, 16, 17, 17, 18, 19, 20, 21, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 24, 25, 25, 25, 25, 25, 26, 26, 27, 27, 28, 28, 29, 29, 29, 29, 29, 30, 30, 31, 31, 31, 32, 32
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(7)-s(3) = 21-3 = 18 = 9*2
s(9)-s(1) = 55-1 = 54 = 9*6
s(10)-s(5) = 89-8 = 81 = 9*9
s(12)-s(5) = 233-8 = 225 = 9*25
s(12)-s(10) = 233-89 = 144 = 9*16
s(13)-s(5) = 377-8 = 369 =9*41

Crossrefs

Cf. A204892, A205873, A205875.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]   (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 9; t = d[c]     (* A205871 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]       (* A205872 *)
Table[j[n], {n, 1, z2}]         (* A205873 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}]   (* A205874 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205875 *)

A205877 Numbers k for which 10 divides s(k)-s(j) for some j

Original entry on oeis.org

6, 7, 9, 11, 12, 12, 15, 16, 16, 17, 17, 18, 18, 19, 19, 21, 21, 21, 22, 22, 22, 23, 24, 24, 24, 25, 25, 25, 26, 26, 27, 27, 27, 27, 28, 29, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 33, 34, 34
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first three terms match these differences:
s(6)-s(3) = 13-3 = 10 = 10*1
s(7)-s(1) = 21-1 = 20 = 10*2
s(9)-s(4) = 55-5 = 50 = 10*5

Crossrefs

Cf. A204892, A205878, A205880.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 600; z2 = 50;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]     (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 10; t = d[c]    (* A205876 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]     (* A205877 *)
Table[j[n], {n, 1, z2}]      (* A205878 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205879 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}]  (* A205880 *)

A205850 [s(k)-s(j)]/4, where the pairs (k,j) are given by A205847 and A205848, and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

1, 3, 2, 5, 4, 2, 8, 13, 22, 21, 19, 17, 34, 58, 57, 55, 53, 36, 94, 93, 91, 89, 72, 36, 152, 144, 246, 233, 399, 398, 396, 394, 377, 341, 305, 644, 610, 1045, 1044, 1042, 1040, 1023, 987, 951, 646, 1691, 1690, 1688, 1686, 1669, 1633, 1597, 1292, 646
Offset: 1

Views

Author

Clark Kimberling, Feb 02 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			The first six terms match these differences:
s(4)-s(1) = 5-1 = 4 = 4*1
s(6)-s(1) = 13-1 = 12= 4*3
s(6)-s(4) = 13-5 = 8 = 4*2
s(7)-s(1) = 21-1 = 20 = 4*5
s(7)-s(4) = 21-5 = 16 = 4*4
s(7)-s(6) = 21-13 = 8 = 4*2

Crossrefs

Cf. A204892, A205845, A205849.

Programs

  • Mathematica
    s[n_] := s[n] = Fibonacci[n + 1]; z1 = 400; z2 = 60;
f[n_] := f[n] = Floor[(-1 + Sqrt[8 n - 7])/2];
Table[s[n], {n, 1, 30}]
u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
Table[u[m], {m, 1, z1}]  (* A204922 *)
v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
d[n_] := d[n] = Delete[w[n], Position[w[n], 0]]
c = 4; t = d[c]    (* A205846 *)
k[n_] := k[n] = Floor[(3 + Sqrt[8 t[[n]] - 1])/2]
j[n_] := j[n] = t[[n]] - f[t][[n]] (f[t[[n]]] + 1)/2
Table[k[n], {n, 1, z2}]    (* A205847 *)
Table[j[n], {n, 1, z2}]    (* A205848 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205849 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205850 *)
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