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 1-5 of 5 results.

A205840 [s(k)-s(j)]/2, where the pairs (k,j) are given by A205837 and A205838.

Original entry on oeis.org

1, 2, 1, 3, 6, 5, 4, 10, 9, 8, 4, 16, 13, 27, 26, 25, 21, 17, 44, 43, 42, 38, 34, 17, 71, 68, 55, 116, 115, 114, 110, 106, 89, 72, 188, 187, 186, 182, 178, 161, 144, 72, 304, 301, 288, 233, 493, 492, 491, 487, 483, 466, 449, 377, 305, 798, 797, 796, 792, 788
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

Let s(n)=F(n+1), where F=A000045 (Fibonacci numbers), so that s=(1,2,3,5,8,13,21,...). If c is a positive integer, there are infinitely many pairs (k,j) such that c divides s(k)-s(j). The set of differences s(k)-s(j) is ordered as a sequence at A204922. Guide to related sequences:
c....k..........j..........s(k)-s(j)....[s(k)-s(j)]/c
2....A205837....A205838....A205839......A205840
3....A205842....A205843....A205844......A205845
4....A205847....A205848....A205849......A205850
5....A205852....A205853....A205854......A205855
6....A205857....A205858....A205859......A205860
7....A205862....A205863....A205864......A205865
8....A205867....A205868....A205869......A205870
9....A205872....A205873....A205874......A205875
10...A205877....A205878....A205879......A205880

Examples

			The first six terms match these differences:
s(3)-s(1) = 3-1 = 2 = 2*1
s(4)-s(1) = 5-1 = 4 = 2*2
s(4)-s(3) = 5-3 = 2 = 2*1
s(5)-s(2) = 8-2 = 6 = 2*3
s(6)-s(1) = 13-1 = 12 = 2*6
s(6)-s(3) = 13-3 = 10 = 2*5

Crossrefs

Cf. A204890, A205587, A205839.

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 = 2; t = d[c]    (* A205556 *)
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}]    (* A205837 *)
Table[j[n], {n, 1, z2}]    (* A205838 *)
Table[s[k[n]] - s[j[n]], {n, 1, z2}] (* A205839 *)
Table[(s[k[n]] - s[j[n]])/c, {n, 1, z2}] (* A205840 *)

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 *)

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 *)

A205843 The least number j such that 3 divides s(k)-s(j), where k(n)=A205842(n) and s(k) denotes the (k+1)-st Fibonacci number.

Original entry on oeis.org

2, 2, 4, 1, 3, 1, 6, 1, 6, 8, 2, 4, 5, 3, 7, 2, 4, 5, 10, 2, 4, 5, 10, 12, 1, 6, 8, 9, 3, 7, 11, 1, 6, 8, 9, 14, 1, 6, 8, 9, 14, 16, 2, 4, 5, 10, 12, 13, 3, 7, 11, 15, 2, 4, 5, 10, 12, 13, 18, 2
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205842.)

Crossrefs

Cf. A204890, A205842, A205845.

Programs

  • Mathematica
    (See the program at A205842.)

A205844 s(k)-s(j), 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

3, 6, 3, 12, 18, 33, 21, 54, 42, 21, 87, 84, 81, 141, 123, 231, 228, 225, 144, 375, 372, 369, 288, 144, 609, 597, 576, 555, 984, 966, 843, 1596, 1584, 1563, 1542, 987, 2583, 2571, 2550, 2529, 1974, 987, 4179, 4176, 4173, 4092, 3948, 3804, 6762
Offset: 1

Views

Author

Clark Kimberling, Feb 01 2012

Keywords

Comments

For a guide to related sequences, see A205840.

Examples

			(See the example at A205842.)

Crossrefs

Cf. A204890, A205842, A205845.

Programs

  • Mathematica
    (See the program at A205842.)
Showing 1-5 of 5 results.

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